text:
Talk:Root of unity/Archive 1

Cyclotomic polynomials
Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 1, −1, or 0; the first polynomial where this occurs is Φ105, since 105=3×5×7 is the first product of three odd primes.

 * It's not immediately clear to me why this property of 105 implies it's the first cyclotomic polynomial with coefficient not equal to 1, -1, or 0. Is this really easy to see, or something I've just forgotten? If it's pretty easy to show, then a quick justification would be good, if it's not very easy to show, I don't think the fact should be mentioned, it leaves the reader dangling.

It's because the pattern for cyclotomic polynomials is more complicated for numbers that are the product of three or more different odd primes and/or their powers (the smallest being, as you've stated, 105) than it is for numbers that are the product of two or fewer. Notice that a power of two doesn't make the polynomial more complicated; this simply means that the coefficients of odd powers of x will have the opposite sign (from -1 to +1 or +1 to -1). Glenn L 23:24, 23 June 2007 (UTC)

n=0
The case $$ n=0 $$ is included in the definition, and it is stated that there are $$ n $$ solutions to $$ z^n=1 $$. Actually the equation $$ z^0=1 $$ has infinitely many solutions, rather than zero. I'll exclude $$ n=0 $$. Bo Jacoby 14:03, 20 September 2005 (UTC)

unitary matrix
"These different roots of unity can be arranged to form the elements of a unitary matrix, and are thus orthogonal to each other. "
 * Are you talking about the n n'th roots of unity, or about the primitive roots of unity ? Please show me how {+i,-i} is arranged into a unitary matrix. Bo Jacoby 13:34, 23 September 2005 (UTC)

That is a standard treatment given in most books on number theory.
 * A detailed exposition of the orthogonality relationship is given in the article character group.

Admittedly, the exposition could be made simpler, with explicit examples for n=2,3,4,5. linas 16:10, 23 September 2005 (UTC)

The characters of a group representation are orthogonal, but the characters of the cyclic group representation are all the roots of unity, not just the primitive ones. I agree with Bo that this must have been a reference to the DFT matrix, and not just to the primitive roots. If something involving just the primitive roots is also possible, please be more explicit. —Steven G. Johnson 18:53, 26 September 2005 (UTC)

 * I never said anything about the primitive roots. Either this article or the article on the character group can/should be expanded to talk about representations of the cyclic group.  I have no idea of what a "DFT matrix" is. Oh, I see, I just followed the link, Discrete Fourier transform and indeed, it does seem to describe the same marix that is given in character group. Not to flaunt my ignorance, I had no idea that DFT had anything to do with his topic; it is not a connection that group theory or number theory people ever bother to mention. linas 23:44, 26 September 2005 (UTC)
 * 1. I made the example of the unitary matrix for n=4 as Linas requested, but Stevenj has removed it.
 * 2. My confusion regarding primitive roots making a unitary matrix is because the text was "these different roots of unity" instead of "the different roots of unity". That has been corrected and the correction is not yet removed by Stevenj.
 * 3. I am pleased that Linas has learned from my insight into the connection between roots of unity and fourier transform. I would like to simplify the articles on fourier transforms using this insight, but I expect that it will be removed by Stevenj. Bo Jacoby 09:26, 27 September 2005 (UTC)
 * The article still links to discrete Fourier transform, which already has liberal mentions of the roots of unity. This connection is well known, and I agree that it should be linked to from this page.  Linking to DFT and discussing connections between concepts has nothing to do with adopting nonstandard notations. —Steven G. Johnson 23:15, 27 September 2005 (UTC)

Notation
I'd be happier is the notation $$1^{\frac{k}{m}}$$ was introduced before it was used. That is a highly non-standard notation; I've never seen anything like that before. I assume its historical in some way? Is there a reference for this notation? linas 16:10, 23 September 2005 (UTC)
 * Hi Linas. You are right of cause, the definition must precede the use. I'll fix it. I have used $$\ 1^x$$ instead of $$\ e^{2\pi i x}$$ for many years as a simple ink-saving notation, but frankly I do not remember from where it came. Perhaps I have invented it myself. I tend to invent things now and then. If $$\ e^{2\pi i }=1$$, then why not write $$\ 1$$ instead of $$\ e^{2\pi i}$$ ? It is really no big deal. It could simplify the confusing notation for fourier transforms. The alternative interpretation, $$\ 1^x=1$$, is uninteresting. Confusion is unlikely to occur because nobody will write $$\ 1^x$$ meaning $$\ 1$$. See exponentiation. Bo Jacoby 10:14, 26 September 2005 (UTC)
 * I've removed this notation. I agree that it is nonstandard.  (I tend to think it is also misleading, because it gives the impression that there is a unique choice of primitive root $$1^{\frac{1}{m}}$$, when in fact this is wholly arbitrary.)  Wikipedia should not be in the business of inventing major new notations from whole cloth.  —Steven G. Johnson 18:51, 26 September 2005 (UTC)
 * $$1^{\frac{1}{m}}$$ is not an arbitrary primitive root, but is equal to $$e^{\frac{2\pi i}{m}}$$ which is well defined. To the readers most of the material in Wikipedia is new, and of cause they don't mind. By deleting material just because is new to you, you prevent Wikipedia from exceeding your level. Think about it. Get wiser. Bo Jacoby 09:07, 27 September 2005 (UTC)
 * Hey, please, be nice. Personal notations are not good here, and that is a solid matter of policy. Charles Matthews 10:52, 27 September 2005 (UTC)
 * I'm sorry. But what is the policy of Wikipedia: that everybody can contribute, or that a censor removes anything that he hasn't seen before simply because he hasn't seen it before ? Bo Jacoby 06:39, 28 September 2005 (UTC)
 * The policy in cases of content disputes is that matters are resolved by discussion, in an atmosphere of mutual respect. The policy in the matter of innovation is that WP is not a place to innovate. Charles Matthews 07:35, 28 September 2005 (UTC)

It can be proved in any number of ways

 * 1) Is is really important to stress that there are more ways to prove it? I think not.
 * 2) Nor do I think that it is true that it can be proved in essentially more than one way. Please provide an alternative proof. Bo Jacoby 10:09, 27 September 2005 (UTC)
 * Proof #2: the roots of unity are eigenvectors of the discretized Laplacian with periodic boundary conditions, which is Hermitian and therefore has orthogonal eigenvectors. Proof #3: multiply the sum S of the roots of unity by one of the roots of unity z; by the rearrangement theorem you get the same sum in a different order; thus, Sz = S and therefore S = 0.   Proof #4: the group theoretic proof already mentioned, based on representation theory.  See also Charles' proofs below.  The existence of the various proofs of the orthogonality of the roots of unity is indeed important, because it means that this orthogonality appears in different ways in many branches of mathematics.  —Steven G. Johnson 23:16, 27 September 2005 (UTC)
 * The proofs: geometric by barycentre of regular n-gon, algebraic by coefficient of xn − 1, complex number by ζS = S — I think these are different ways. By the way, the point is that we may include material from books, but don't have to. Charles Matthews 15:00, 27 September 2005 (UTC)

Thank you very much for these alternative proofs. They clarify the claim that there are several proofs. Without that clarification the claim is confusing to the reader. Can the set of proofs of a mathematical statement be counted ? From the point of view of Gödel a countable infinity of proofs can be trivially generated from a single proof, but that is cheating. That makes the claim that there are several proofs meaningless, because it applies for any provable statement. So that is not what is meant. The 'different' proofs must be somehow logically independent. Are the above proofs logically independent ? Can the set of logically independent proofs of a provable mathematical statement be counted ? I don't know the answer. Does anybody ? I do not think that these thoughts are relevant to the subject of summing the roots of unity. So I still don't think that the number of ways in which it can be proved should be mentioned in the article. Bo Jacoby 09:32, 29 September 2005 (UTC)

 * Well, you can cut it out. 'Logical independence' of proofs isn't a useful idea, I think. Most people recognise the idea that there can be proofs from different starting points. You are wrong about it being irrelevant. WP's articles aim for completeness, so certainly mentioning how one topic is seen in different fields is relevant. Charles Matthews 13:09, 29 September 2005 (UTC)

What is the definition of $$\omega_n$$
"The primitive root $$e^{2 \pi i /n}$$ (or its conjugate) is often denoted by $$\omega_n$$, especially in the context of discrete Fourier transforms."
 * 1) What is the definition of $$\omega_n$$ ?    $$e^{2 \pi i /n}$$ or $$e^{-2 \pi i /n}$$ ?
 * It could be either—there is no universal definition, as far as I can tell. Hence the or in parentheses. —Steven G. Johnson

 * 1) Do you want to teach people bad mathematics just because it is often found in the books, or do you want to present quality mathematics ? Bo Jacoby 10:09, 27 September 2005 (UTC)
 * Wikipedia does not present original research. If you think that the mathematics in standard textbooks is bad, then you should publish elsewhere and convince people.  Wikipedia is not the place for promulgation of new standards. —Steven G. Johnson

 * WP is a reference, not a textbook. Readers might be using WP to look up something forgotten, or to get a more complete idea of a subject that is already familiar.

 * Thus, among other things, an article should discuss alternate notations that are in common use, so that when the reader compares their textbook to the WP article, they find them to be consistent. The few readers who might be trying to learn something completely new should be directed to textbook references.
 * If the 1^{k/n} notation is indeed in "common use", then we should indeed give it. However, you've given zero evidence of this. —Steven G. Johnson 23:11, 27 September 2005 (UTC)

 * As to Steven's edits, perhaps they were too drastic. Please try to find a middle ground. linas 13:30, 27 September 2005 (UTC)

Notation, again
Bo,

Please stop using the $$1^{k\over n}$$ notation until such time as you can provide a book reference for this notation. After the discussion above, I am surprised that you put it back in again. linas 20:33, 30 September 2005 (UTC)
 * Well, Linas, I didn't find the discussion decisive. You requested a compromise, and I considered "the notation is convenient though nonstandard" a compromise. Wasn't PAR too in favour for the notatation? Bo Jacoby 06:32, 3 October 2005 (UTC)
 * It's not a standard notation. It doesn't belong on WP, under policy. Charles Matthews 07:00, 3 October 2005 (UTC)

n=0
Now please explain to me why the case n=0 is included when in fact it is not defined? My change of $$\mathbb{N}_0$$ to $$\mathbb{N}$$ was revoked. Bo Jacoby 06:32, 3 October 2005 (UTC)

 * Bo, no one has any problem with minor clarifications like this. The problem is that you mix them in with large-scale changes of notation in the article, which are hard to undo except by reversion...then your littler changes get lost in the shuffle.  It also doesn't help that you react with moral outrage to every perceived slight.  —Steven G. Johnson 05:09, 6 October 2005 (UTC)

Thank you. I think your last edit changed $$\mathbb{N}$$ into $$\mathbb{N}_0$$. It should have been the other way round. $$\mathbb{N}$$ excludes 0 and is correct. $$\mathbb{N}_0$$ includes 0 and is incorrect.

Hermitian matrix
Steven,

You added the text
 * The roots of unity also appear as the eigenvectors of a Hermitian matrix (a discretized one-dimensional Laplacian with periodic boundaries), from which the orthogonality properties also follow.

Can you give an explicit form for this matrix? I am left to guess what it might be. Is it the tridiagonal matrix with -1 2 -1 along the diagonal? Different readers will no doubt make other guesses. linas 20:33, 30 September 2005 (UTC)

 * Almost... -1 2 -1 would be -∇2, whereas +∇2 would be 1 -2 1 (except for the boundary elements which have to wrap around). Not that it matters; an overall sign flip only affects the eigenvalues and not the eigenvectors, and a negative Laplacian is actually more natural in the physical context of a wave equation.  (I guess a more precise description would be the center-difference discretized Laplacian.)  It's discussed in this article by Strang:
 * The Discrete Cosine Transform
 * I'll add a reference to the article. —Steven G. Johnson 01:23, 1 October 2005 (UTC)

 * FYI, the discretized Laplacian is not, of course, the only Hermitian matrix that has the roots of unity as eigenvectors. Any discretized Hermitian operator with discrete translational symmetry under cyclic shifts will do. This follows, once again, from representation theory, because the eigenvectors of a Hermitian operator can always be chosen to transform as one of the representations of its symmetry group, which in this case is a cyclic group...as usual, everything is connected. For the same reason, it actually doesn't matter what discretization of the Laplacian you use, as long as it is symmetric (in order to get a Hermitian matrix) and periodic. —Steven G. Johnson 01:34, 1 October 2005 (UTC)

orthogonality
Steven. The $$n$$th roots of unity can be arranged to form an $$n \times n$$ matrix whose $$(j,k)$$th entry is
 * $$U_{j,k}=n^{-\frac{1}{2}} \omega_n^{jk}$$

is a lie. The elements of the matrix are not the roots of unity. I keep correcting it and you keep repeating it. Please argue in the discussion forum before you insist on removing my edits. Bo Jacoby 08:55, 5 October 2005 (UTC)

 * Um, the article doesn't state that the entries of the matrix are roots of unity. It clearly gives the sqrt(n) factor, and the matrix is indeed based on an arrangement of the roots of unity.  If you want to be pedantic about the word "arranged" (even though the equation makes things perfectly clear, whereas English is more flexible than you seem to think), fine, I replaced it with "used".  Throwing around accusations of "lies" only makes you look foolish.  —Steven G. Johnson 17:07, 5 October 2005 (UTC)

 * By the way, your edit was reverted not because of the minor changes in the text, but because of an unhelpful wholesale change of notation. This was clearly explained in my edit comments.  —Steven G. Johnson

The reader has bigger difficulties than the author in understanding the text. So the author should lend ear to the reader's comments. As a reader I changed several formulations which were simply wrong, such as the case n=0. But you reverted everything - not merely what you disagreed on, but everything. Do you realize know how many times you have done that? Please stop it. It is a backwards step instead of a forward step. I edited for a reason. A word that confuses rather than clarifies should be removed. The word arrange is explained in the Longman Dictionary of Contemporary English: to put into a correct, pleasing or desired order. The examples given are: To arrange flowers in a vase, and: The books are arranged on the shelves in alphabetical order. Please don't argue that it can ever mean divide by the square root of the number of items, and then copy them to fit into a square matrix. Any reader is pedantic about words until he gives up reading. Please don't argue ad hominem that I am specially pedantic and therefore should be taken less seriously. I would not be this foolishly indiplomatic if I had seen diplomacy work. But to your surprise I wish to say that I do enjoy working with you in improving WP. It is fun and it is worth while. Let's be friends. Bo Jacoby 09:33, 6 October 2005 (UTC)

 * Bo, I apologize for any ad hominem comments. On your part, please avoid charged language like "lies" etcetera.  As I mentioned above, the reason for the reverts were because you mix your small corrections in with wholesale changes of notation that I could not agree with, and the only practical way to undo such a large change is a revert.  I've tried to scan through the diff to see if you made other substantive changes that I should try to keep, but it is easy to miss little changes of wording.  —Steven G. Johnson 15:40, 6 October 2005 (UTC)
 * (PS. If you arrange flowers in a vase, that doesn't mean that you haven't put anything else into the vase. That doesn't mean I mind changing the wording, but I do think you overreacted. —Steven G. Johnson)

I agree that my charged language make me look like a fool and I am not happy about it. I apologize. Bo Jacoby 12:26, 7 October 2005 (UTC)

Lamest edit wars ever
Not quite there, but certainly this page is heading in the direction ... Charles Matthews 20:07, 5 October 2005 (UTC)

Geometrical Representation of Roots
It was mentioned that "It can be shown that they are located on the unit circle of the complex plane and that in that plane they form the vertices of a n-sided regular polygon with one vertex on 1.". Could somebody please show it? Thanks a lot.
 * If x is an n th root of unity, then xn=1. Take the complex conjugate of that equation to see that the conjugate x* is a root of unity: (x*)n=1. Multiply these two equations to see that the positive real number xx*=|x|2 is a root of unity. So xx*=1, because 1 is the only positive real root of unity. So |x|=1. That means that x is on the unit circle. This was the first part of the assertion. Now consider the regular n-sided polygon with center in 0 and one of the vertices in 1. Consider the first vertex A in clockwise direction from 1. The triangle (0;1;A) is similar to triangle (0;A;A2) according to Complex_number. So A2 is the 2'nd vertex on the polygon. And (by induction) Ak is the kth vertex on the polygon. The numbers A, A2, ..., An=1. are n'th roots of unity if A is. This was the second part of the assertion. Bo Jacoby 15:29, 30 November 2005 (UTC)
 * I suspect "show it" may have been a request for an illustration, although the proof might be good to add too. Doops | talk 18:31, 30 November 2005 (UTC)

Cyclotomic polynomials, incorrect statement

 * Applying Möbius inversion to the formula gives
 * $$\Phi_n(z)=\prod_{d\,\mid n}(z^{n/d}-1)^{\mu(d)},$$
 * where μ is the Möbius function.

The Möbius inversion is about sums, not about products. And the cyclotomic polynomial is irreducible and thus cannot be written as a product of polynomials in a non-trivial way. I will remove the sentence from the article. Bo Jacoby 11:26, 21 June 2006 (UTC)

 * I am afraid that change is not valid: the expression given, and the reason (Mobius inversion) are both correct - see page 206 of "Algebraic Number Theory" by Frohlich and Taylor. I will re-instate it. Madmath789 11:54, 21 June 2006 (UTC)

Example. n = 4. The controversial formula gives:
 * Φ4(z) = (z1−1)μ(4)(z2−1)μ(2)(z4−1)μ(1) = (z1−1)0(z2−1)−1(z4−1)1.

UPS !
 * Φ4(z) = z2+1

You are right and I was wrong. Thank you, and apology. Could you please clarify the connection to the Möbius inversion which is about sums? Bo Jacoby 12:44, 21 June 2006 (UTC)

 * No problem - what I have done is added a very short "generalisations" section to the Möbius inversion article, which states the multiplicative version. Madmath789 12:53, 21 June 2006 (UTC)

 * Incidentally, care is also needed with the statement "the cyclotomic polynomial is irreducible", as for example (5th cyclotomic polynomial:
 * $$x^4+x^3+x^2+x+1 = (x^2+\frac{-1+\sqrt{5}}{2}x+1)(x^2+\frac{-1-\sqrt{5}}{2}x+1).$$ So $$\Phi_5(x)$$ is reducible over $$\mathbb{Q}(\sqrt{5})\mbox{ but irreducible over }\mathbb{Q}.$$

$$\mbox{If } F(n) = \prod_{d|n} f(d),\mbox{ then } f(n) = \prod_{d|n} F(n/d)^{\mu(n/d)}.$$ should be $$\mbox{If } F(n) = \prod_{d|n} f(d),\mbox{ then } f(n) = \prod_{d|n} F(n/d)^{\mu(d)}.$$ Agree? Bo Jacoby 15:13, 21 June 2006 (UTC)

 * Yes - either

$$\mbox{If } F(n) = \prod_{d|n} f(d),\mbox{ then } f(n) = \prod_{d|n} F(n/d)^{\mu(d)},$$ or $$\mbox{If } F(n) = \prod_{d|n} f(d),\mbox{ then } f(n) = \prod_{d|n} F(d)^{\mu(n/d)},$$ but not the version I stated - I will correct it, thanks. Madmath789 15:29, 21 June 2006 (UTC)

There is a small mistake in the calculation - a typo I'm guessing - of the 6th cyclotomic polynomial. The term (z-1) in the product should be to the 1st power not to the -1 power as the mobius function maps 6 to 1 not -1. This would lead to a prdouct giving a polynomial of degree 2 as desired. 71.194.89.197 01:27, 20 March 2007 (UTC)

 * Indeed, you are right. Fixed now. -- EJ 12:58, 20 March 2007 (UTC)

So I checked this page and there is still a -1 power right next to the (z-1) term in the calculation. I thought I would be proactive nad look to see if I could edit it myself but it appears that it is edited correctly (as you say) but the display shows something different, what gives? 71.194.89.197 03:44, 21 March 2007 (UTC)

Convergent Points
I think a section mentioning that $$z^{\infty} = 1$$ for all points on the unit circle. But it should also mention those points which are not finite roots of unity. That is, the theoretical "infiniteth" roots of unity, whos infinite exponentation converge to one, but no finite exponentation there of then actually reaches one. It can be shown that these points distance along the unit circle (Not lateral distance. Distance on the circle) is pi divided by some irrational number iff it is a primitive infiniteth root. (Because the distance will for nth roots of unit for finite n is pi, the circumference of the unit circle, divided by n, times k, for all natural k less than n.) But I don't think infinitethis the right word to use in article. Anyone with better terminology, ideas, or objections to this going in the article? -- He Who Is[ Talk ] 12:20, 23 June 2006 (UTC)

No, there should be no infinity in the article. It is purely algebraic.

By the way, the only solution to
 * $$\lim_{n\rarr\infty}z^n=1$$

is
 * $$\ z=1$$

so there is no nontrivial infinite'th root of unity.

The number
 * $$\ e^{2\pi i/x}$$

where x is irrational, might be called an irrational root of unity. Perhaps that is what you are talking about. Bo Jacoby 08:06, 26 June 2006 (UTC)

Well the kth nth root of unity (Let's assume the nth roots of unity are ordered by the value of k in the following formula for the moment) we know equals $$e^{2\pi ik/n}$$ for all natural k less than n. The distance from 1 to this number along the unit circle will be irrational for all kth nth root of unity where k and n are integers, because pi is irrational.but what about if $$\frac{k}{n}$$ is irrational? Is it still a root of unity? For some "irrational root of unity" z, as you put it, no finite power could bring it to one. So I would say that it only stands to reason that it in fact can be shown that:

 * $$\lim_{n\to\infty} z^n = 1 $$

Though I may be wrong. I haven't really given this much thought. -- He Who Is[ Talk ] 17:13, 26 June 2006 (UTC)

 * If $$z = e^{2\pi ia},$$ with a irrational, then the sequence $$z^n$$ cannot converge to 1, and in fact, (if I remember correctly) the values $$\{z^n\}$$ are actually dense in the unit circle. Madmath789 21:05, 26 June 2006 (UTC)

So then only kπ for rational k represent the distance on the unit circle from 1 to a root of unity? Or did I not understand corrrectly? -- He Who Is[ Talk ] 21:36, 26 June 2006 (UTC)

 * Yes, that is how it is - rational k gives an nth root of unity and irrational k do not. Madmath789 21:54, 26 June 2006 (UTC)

Notation
A common notation for $$e^{2 \pi i /n}$$ I've noticed in many books is $$\zeta_n$$ or just $$\zeta$$. Should this be mentioned here or is it rarer notation than I think? I daren't edit this page for fear of starting a lame edit war :) Moxmalin 22:44, 1 April 2007 (UTC)

 * Edit wars are made by disrespectful editors who do not follow the rules: Help:Reverting, not by those who in good faith improve the articles. Be bold and edit. Bo Jacoby 09:44, 2 April 2007 (UTC).

Cyclotomic polynomials
Quote:
 * While in general polynomials of degree higher than 4 cannot be solved by radicals, cyclotomic polynomials can. This is because their Galois group is abelian, and hence solvable. Thus, every root of unity has an expression in radical form. Algorithms exist for calculating such expressions.

Trivially every root of unity has an expression in radical form: $$\sqrt[n]1$$. What is the nontrivial content of the quote? Bo Jacoby 20:04, 9 April 2007 (UTC).

 * The only possibility I can think of is that the quote was trying to say that the real and imaginary parts of roots of unity can be expressed by real radical expressions. However, even if that is true (which I do not know), the first two sentences still make no (nontrivial) sense, the quote needs a reformulation and a solid reference. I'm cutting it out for the moment. -- EJ 12:14, 10 April 2007 (UTC)

 * Aha, I'm talking nonsense. Let p be an irreducible integer polynomial of degree 3 with 3 distinct real roots, whose discriminant has a rational square root (e.g., $$x^3-3x-1$$). Then roots of p cannot be expressed using real radicals ("casus irreducibilis" ). OTOH, its splitting field is an Abelian extension of $$\mathbb Q$$ (being of degree 3), hence it is contained in some $$\mathbb Q(\zeta_n)$$ by the Kronecker-Weber theorem. Thus, $$\Re\zeta_n$$ and $$\Im\zeta_n$$ cannot be expressed using real radicals either. -- EJ 10:21, 11 April 2007 (UTC)

 * I've been away for a long time, so I haven't responded. But well, basically, all that I wanted to convey is the content of . I'm not really sure what the best way of putting it is - the issue is that the fact that the 3rd root of unity is 1/2(1 + i * root 3) (and not some unwritedownable transcendental value, say) is not a fluke, and that for any given n, it is possible to construct some finite combination of integers, elementary operations, i, and the real kth root to give the root of unity exactly. I don't really understand what you mean by it being impossible, since well, we have examples of it being true, and an algorithm to find such radicals. (The article attributes the proof to Gauss)--Fangz 20:35, 20 June 2007 (UTC)

 * By being impossible I mean that it is literally impossible: there exists an n such that the real and imaginary part of the primitive nth root of unity cannot be expressed using rational operations and real roots. If you do not understand the proof of this fact above, I can explain it in more detail, just ask. Notice that this does not contradict the Weber-Keckeisen paper: the radical expression they construct do involve roots of nonreal complex numbers, as can be seen from the examples.

 * The problem is that it is totally unclear what the paper is trying to accomplish. They do not give their working definition of "radical expression", and the standard definition as used in algebra textbooks allows to adjoin any radical of a previously constructed number: then roots of unity are trivially expressible as, well, roots of unity.

 * I strongly believe that no information is better than false or incomprehensible information. The second paragraph you included is blatantly false. The first one was originally either trivial or it had unclear and unexplained meaning, and after your modification, it is also false. I will thus remove it again.

 * I do not have access to Disquisitiones Arithmeticae; it would be most wonderful if somebody who can looked it up to find the definition of "radical expression" it really uses, instead of trying to guess it, which just proved not to work. -- EJ 10:03, 21 June 2007 (UTC)

 * I forgot to say: the explanation using commutativity (hence solvability) of the Galois group of the cyclotomic extensions is bogus at any rate: the usual proofs of solvability in radicals of solvable Galois extensions begin with something to the effect of "roots of unity are radicals, hence we may assume w.l.o.g. that the base field contains all roots of unity". Obviously, this makes the explanation circular. -- EJ 10:27, 21 June 2007 (UTC)

By backtracking the references in, I found at depth 2 the paper , which gives a coherent definition of radical expressions. Basically, it goes as follows: an expression involving rational operations and radicals solves an irreducible polynomial equation p=0, if every choice of values for the radicals gives a root of p. (Apparently, the expressions have to be dag-like for this to work, but so be it.) In other words, all possible values of the expression under various choices of the radicals have to be conjugate.

This condition rules out the trivial expression $$\zeta_n=\sqrt[n]1$$, it is satisfied by the usual expressions in Root of unity (modulo the daglikeness caveat), and overall it seems to make sense. So I think this is it. There is no guarantee that Weber and Keckeisen actually use this definition of radical expressions, but at least it sounds plausible, so I'll give it a try. -- EJ 10:06, 22 June 2007 (UTC)

pedantry
Yes, it's pedantic -- I know; but as far as I know the convential definition requires unities to coincide (i.e. considers selection of unity to be an operation). In any case, the interesting case is when the unity is stable. --VKokielov 18:39, 4 June 2007 (UTC)

My addition
Because the roots of unity are algebraic, and because they are given by the trigonometric functions sine and cosine, it follows that the sine (or cosine) of any angle which is a rational fraction of the revolution 2π must be computable by a finite number of additions, multiplications, and square roots.

Even if I'm right, I don't want to stick it in out of context. --VKokielov 17:12, 5 June 2007 (UTC)
 * Square roots!.. I'm thinking of powers of two.  If the root isn't a power of two, then these needn't be square roots at all.  --VKokielov 17:39, 5 June 2007 (UTC)
 * I think that is true, but the fact that the numbers are algebraic is not enough - since not all algebraic integers are expressible by a finite number of additions, multiplications, and roots. (See Abel–Ruffini theorem). Fortunately, however, in the case of roots of unity, they can, (see referenced article above), but again, this is a property beyond that of just being algebraic.--Fangz 20:49, 20 June 2007 (UTC)
 * This seems to be disproven by EJ's argument above (assuming its correctness, which I'm not qualified to check). Tesseran 06:16, 28 July 2007 (UTC)

Hermitian matrices
Quote from the subsection on Orthogonality: "The roots of unity appear as the eigenvectors of Hermitian matrices". I think this is incorrect and perhaps should be: "The roots of unity appear as the eigenvectors of Unitary matrices". The eigenvalues of Hermitian matrices are real numbers. Bo Jacoby 10:38, 1 August 2007 (UTC).

 * No, it's correct as-is, although a little vague. If one constructs any translation-invariant Hermitian matrix (invariant under cyclic shifts), e.g. a discrete Laplacian with periodic boundaries (as in the Strang reference), the eigenvectors are (or can be chosen to be) the roots of unity (i.e. it is diagonalized by the DFT).  This is a consequence of group representation theory, and is essentially an instance of Bloch's theorem.  (Actually, I don't think the matrix has to be Hermitian, although this was Strang's example.  It just has to be non-defective and translation-invariant.  However, from a Hermitian matrix like the discrete Laplacian where the eigenvalues are non-degenerate, it is easier to see orthogonality as a consequence.)  The eigenvalues of a Hermitian matrix are real. The eigenvectors are not in general. —Steven G. Johnson 17:25, 1 August 2007 (UTC)

Thanks. The matter should either be clearly explained or completely omitted. ("Was sich überhaupt sagen läßt, läßt sich klar sagen; und wovon man nicht reden kann, darüber muß man schweigen" says Wittgenstein). Bo Jacoby 20:13, 1 August 2007 (UTC).

 * As long as there is a reference for where to go for more information, brief mentions of more advanced topics are fine, and in fact desirable. Encyclopedias, by their nature, should be encyclopedic, and have a very different goal from, say, a minimalist axiomatic development. —Steven G. Johnson 22:55, 1 August 2007 (UTC)

Some readers find the articles technical and confusing. Even brief mentions should be clearly written. It was suggested by BobK in Talk:Discrete_Fourier_transform that two articles be written, basicly one in your style and one in mine. You had the DFT-article on your own; (you reverted all my edits, violating WP:DR). Now I suggest that you leave this elementary article for me to edit for a while. Your writing is technical and confusing to many readers. Bo Jacoby 05:53, 2 August 2007 (UTC).

We have other pages on Fourier analysis
The specific application of roots of unity to Fourier analysis is described in detail by discrete Fourier transform, and we also have specific articles on aliasing etc. This page should refer to those pages, but there is no reason to have a separate description of computing Fourier components, aliasing, etc. in this page. Hence I am removing that information. (Note that the longish discussion in the current "computing Fourier components" is merely a restatement of the fact that the matrix is unitary, and is especially redundant.) —Steven G. Johnson 15:57, 7 August 2007 (UTC)

Examples
I restored the examples because I believe they communicate the mechanics of basic computing with roots of unity and cyclotomic polynomials. Others may differ, of course, which is why I started this thread on the talk page. -Zahlentheorie (talk) 21:48, 26 December 2007 (UTC)

Seventeenth root of unity
Is it worth putting the following example (extracted from ) of a seventeenth root of unity?

$$ x + i\sqrt{1-x^2}$$

where

$$ x = \frac{ -1 + \sqrt{17} + \sqrt{2(17 - \sqrt{17})} + 2 \sqrt{17 + 3\sqrt{17} - \sqrt{2(17-\sqrt{17})} - 2\sqrt{2(17 + \sqrt{17})}} }{16} $$

I haven't verified that this is correct, or that it is the simplest form possible. It may be worth mentioning because it only involves taking square roots, which is due to a famous result of Gauss (on the constructibility of regular polygons by ruler and compass). Perhaps also interesting is that the square roots being taken here are positive square roots of positive real numbers. —Preceding unsigned comment added by DRLB (talk • contribs) 21:39, 4 January 2008 (UTC)
 * See the formula in the article Heptadecagon. 03:38, 5 January 2008 (UTC). —Preceding unsigned comment added by Bo Jacoby (talk • contribs)

Missing reference?
Twice in this article, the following statement occurs:
 * This problem was discussed on the newsgroup es.ciencia.matematicas, and the article is here.

However the link given is "dead" (since the #External links section no longer exists), and further there doesn't seem to be anything left under the "References" or "See also" sections which match up with the es.ciencia.matematicas reference. I scanned a page's worth of edits in the History, but noticed no obvious references to removing this reference in any of the edit summaries. Does anyone more familiar with this article have any thoughts on where this reference went? --PeruvianLlama(spit) 04:28, 5 March 2008 (UTC)

 * In general, this section doesn't really belong in the article. Why is this problem that happened to be discussed on some newsgroup notable enough to go in the encyclopedia article on roots of unity, and if it is so notable why isn't there a published reference for it in a reputable source?  Hence I have removed it and include it here for the benefit of discussion.  —Steven G. Johnson (talk) 23:23, 14 March 2008 (UTC)

Extracting coefficients
As a special case of orthogonality, we have

 * $$\sum_{k=0}^{n-1} \left(e^{2 \pi i k/n}\right)^j =

\begin{cases} n \quad \mbox{if} \quad j \equiv 0 \mod n \\ 0 \quad \mbox{otherwise}, \end{cases}$$

i.e. the sum of the roots of unity raised to some fixed power $$j$$ is $$n$$ if $$j$$ is divisible by $$n$$, and zero otherwise.

This implies the following useful fact. If we have a polynomial or a series $$f(x)$$ in $$x$$, where
 * $$f(x) = \sum_j f_j x^j,\,$$

then
 * $$\frac{1}{n}

\sum_{k=0}^{n-1} f\left(e^{2 \pi i k/n} \, x \right) = \sum_{n|j} f_j x^j.$$

As a particular case, we have
 * $$\frac{1}{n}

\sum_{k=0}^{n-1} f\left(e^{2 \pi i k/n} \right) = \sum_{n|j} f_j,$$ i.e. the sum of the coefficients that are divisible by $$n$$, assuming it exists.

As an example as to how this may be applied, ask the following question: suppose we choose an integer at random from $$[0, 10^{11}-1].\,$$ What is the probability that its digit sum will be divisible by $$11$$?

Clearly the generating function of the digit sums of the integers in $$[0, 10^{11}-1]\,$$ is
 * $$ f(x) = \left(x^0 + x^1 + x^2 + \cdots + x^8 + x^9\right)^{11}.$$

We seek the sum of the coefficients of $$x$$ raised to a power that is divisible by $$11$$, giving
 * $$\frac{1}{11}

\sum_{k=0}^{10} f\left(e^{2 \pi i k/11} \right) = \frac{1}{11} f(1) + \frac{1}{11} \sum_{k=1}^{10} \left(- \left(e^{2 \pi i k/11}\right)^{10} \right)^{11},$$ because
 * $$x^0 + x^1 + x^2 + \cdots + x^8 + x^9 = - x^{10}$$

when $$x$$ is an eleventh root of unity that is not one. Additional simplification now yields

\frac{1}{11} 10^{11} - \frac{1}{11} \sum_{k=1}^{10} \left(e^{2 \pi i k}\right)^{10} = \frac{1}{11} 10^{11} - \frac{1}{11} 10.$$ This means that the desired probability is
 * $$ \frac{1}{11} - \frac{1}{11 \times 10^{10}} =

\frac{909090909}{10000000000} \approx \frac{1}{11}.$$

This problem was discussed on the newsgroup es.ciencia.matematicas, and the article is here.

The next-to-leading coefficient
We have the following equality:
 * $$ \left[z^{\varphi(n)-1}\right] \Phi_n(z) = - \mu(n)$$

where $$\mu$$ is the Moebius function and $$[z^n]$$ is the coefficient-extraction operator for formal power series.

To see this, observe that
 * $$ \left[z^{\varphi(n)-1}\right] \Phi_n(z) = - \sum_{(k, n)=1} z_{k, n},$$

where the $$z_{k, n}$$ are the primitive roots of unity, so we must show that
 * $$ \mu(n) = \sum_{(k, n)=1} z_{k, n},\,$$

which we do by complete mathematical induction.

It certainly holds for $$n=1$$ and $$n=2$$, because
 * $$ \mu(1) = 1 = z_{1, 1} \quad \mbox{and} \quad \mu(2) = -1 = z_{1, 2}.$$

Now suppose it holds for $$m1} \sum_{(k, n)=d} z_{k, n}\,.$$

But for $$d>1\,$$ we have
 * $$\sum_{(k, n)=d} z_{k, n} = \sum_{(l, n/d)=1} z_{l, n/d} = \mu(n/d)\,$$

where the last equality is by induction (note that $$d>1\,$$ implies $$n/d1} \mu(n/d)$$

which is
 * $$\mu(n) - \sum_{d|n} \mu(n/d) = \mu(n) - \sum_{d|n} \mu(d) = \mu(n),\,$$

because the sum of the Moebius function evaluated at the divisors of an integer $$n$$ is zero. QED.

This problem was discussed on the newsgroup es.ciencia.matematicas, and the article is here.

Needs re-writing
--Tangi-tamma (talk) 01:28, 6 April 2008 (UTC)
 * who will read this busy stuff. Remove all the heavy stuff - try to do with examples.

Notation
I wrote a subsection on notation, but User:Stevenj reverted it immediately. (He is allergic to 11/n, no matter whether it means 1 or e2πi/n, and he shows no respect for the rules of WP:Reverting). Somebody else please write a subsection on notation. Bo Jacoby (talk) 11:16, 20 October 2008 (UTC).

 * You have not been able to cite a single reputable source that uses the 11/n notation (i.e., $$\sqrt{1}=-1$$) for primitive n-th roots of unity. As has been repeatedly pointed out to you, your repeated attempts to promote this notation of yours on Wikipedia is against policy.  Hence, it is perfectly justifiable to revert it on sight when you periodically try to sneak it back in.  —Steven G. Johnson (talk) 15:56, 20 October 2008 (UTC)

ith root of 1
I can't find an explicit calculation of the ith root of 1. It's not at Exponentiation, either. FWIW, given that $1/i$ = –i, then 1($1/i$) is equivalent to 1–i, and thus to $1/1^{i}$ as well. — Loadmaster (talk) 15:57, 4 March 2009 (UTC)
 * $$e^{\frac{2\pi i}{i}}=e^{2\pi }\approx 535.4916555247\ldots $$ --84.228.133.241 (talk) 12:54, 29 October 2009 (UTC)
 * Complex exponentiation is multivalued: $$1^{1/i}=e^{2\pi ni/i}=e^{2\pi n}\,$$ for any integer n. Note that this is exactly the same set of values as for 1i.—Emil J. 12:33, 29 March 2010 (UTC)

 * ... because $$(1^i)^i=1^{(i^2)}=1^{-1}=1$$ Gandalf61 (talk) 12:39, 29 March 2010 (UTC)

Untitled comment
There is a mistake in the examples section it says the negative of a root is always also a root and gives the example: (-1/2+i Sqrt(3)/2) and (1/2+i Sqrt(3)/2) as the two primitive cube roots of 1, however, it should read that the root and its Complex Conjugate are always roots because (-1/2+i Sqrt(3)/2)^2 is the other primitive root and this is in fact (1/2-i Sqrt(3)/2). --Kyle —Preceding unsigned comment added by 99.48.65.242 (talk) 22:13, 19 August 2010 (UTC)

 * It doesn't say that. The article was correct. RobHar (talk) 23:07, 19 August 2010 (UTC)

Summation Section

 * $$\sum_{k=0}^{n-1} z^k = \frac{z^n - 1}{z - 1} = 0 .$$

How does that equal 0?

For example, if Z is 20 and n is 2. Then 399/19 = 21, not 0. Also $$ z^{n} - 1 = (z-1) * sum_{k=0}^{n-1} z^k$$ is much better looking.

 * It equals zero because z is assumed to be an nth root of unity, hence zn = 1. (Of course, 20 is not a root of unity). RobHar (talk) 05:02, 22 August 2010 (UTC)

Also, shouldn't there be a QED or some proof termination signal after $$\sum_{k=0}^{n-1} z^k = \frac{z^n - 1}{z - 1} = 0 .$$? Because otherwise the rest of the section seems like a continuation of the proof, and it's not clear. 136.152.167.55 (talk) 00:13, 31 January 2012 (UTC)

radicals
I removed the sentence
 * In general, the real and imaginary parts are given by expressions involving nested radicals and are therefore irrational numbers.

from the article as it is either extremely misleading or downright false depending on the reading.

First, "are therefore irrational numbers" is non sequitur, that something is expressible by an expression involving nested radicals does not imply that it cannot be expressed as a rational in another way. There are roots of unity whose real or imaginary part is rational, such as n = 6.

Second, what are the "expressions involving nested radicals"? —Emil J. 12:26, 23 August 2010 (UTC)
 * If the radicals are meant to be complex, then the first part of the statement is trivial: an nth root of unity can be expressed as $$\sqrt[n]1$$. Moreover, then it is terribly misleading to speak of the real and imaginary parts, since the root itself is easier to express than its two coordinates.
 * If the radicals are meant to be real, then the statement is false in general. The real and imaginary parts of the primitive nth root of unity can be expressed using real radicals iff they can be expressed using square roots iff n is a product of pairwise distinct Fermat primes and a power of 2. (The first equivalence follows from casus irreducibilis.)

Thanks, Emil, you've helped me clarify my thinking.

Theorem: If both the real and imaginary parts of z are rational, and z is a root of unity, then z is 1 (resp −1, ±i ) and it is a primitive first (resp second, fourth) root.

Proof: Let $$z=\frac{a}{b} + i\frac{c}{d}.\;$$ Write the fractions with a common denominator: $$z=\frac{e}{m} + i\frac{f}{m} = \frac{e + i f}{m}.\;$$ Both the numerator and denominator are Gaussian integers, and have a unique factorization into a unit times a product of Gaussian primes: $$z=\frac{up_1p_2\dots}{vq_1q_2\dots}.\;$$

Now if the kth power of z is one we have $$\frac{u^k}{v^k}p_1^kp_2^k\dots=q_1^kq_2^k\dots,\;$$ and by unique factorization the ps and qs cancel, leaving a power of a unit = 1. QED

(Assuming this is a valid proof), I guess it counts as "original research", although that seems like a stretch. I am sure there is a textbook somewhere with this as an exercise, but I don't actually know of one. Is it OK to state in the article something along the lines of "using the properties of Gaussian integers it is a straightforward exercise to show ..." and putting in a footnote outlining the above proof?

BTW, what is the justification for the rule that nth is better than nth? I think the latter is much easier to read.

Virginia-American (talk) 05:13, 24 August 2010 (UTC)
 * You are responding to something I didn't question, but anyway: the proof is valid as far as I can see, but it is clearly way behind the lines of WP:OR. The mere idea that it calls for a proof in a footnote should tell you that it actually requires a source. (As an aside: it's also easy to enumerate roots of unity whose real or imaginary part is rational. If z = a + ib is a primitive nth root of unity and a is rational, then $$\mathbb Q(z)=\mathbb Q(\sqrt{a^2-1})$$ is an (at most) quadratic extension. Since $$[\mathbb Q(z):\mathbb Q]=\varphi(n)$$, we have $$\varphi(n)\le2$$, hence n = 1, 2, 3, 4, or 6. If, on the other hand, b is rational, then $$\mathbb Q(z)\subseteq\mathbb Q(\sqrt{1-b^2},i)$$. This is either a quadratic extension, leading to the numbers above, or it is an extension of degree 4 whose Galois group is the Klein four-group. If $$n=\prod_ip_i^{e_i}$$ is a prime factorization, then $$\operatorname{Gal}(\mathbb Q(z)/\mathbb Q)\simeq(\mathbb Z/n\mathbb Z)^\times\simeq\prod_i(\mathbb Z/p_i^{e_i}\mathbb Z)^\times$$, and $$(\mathbb Z/p_i^{e_i}\mathbb Z)^\times$$ is cyclic except when pi = 2 and ei ≥ 3, when it is a product of two cyclic groups, hence we get n = 8, 12 as the only other possibilities. Some of these turn out not to have rational imaginary part, it actually only holds for n = 1, 2, 4, 12.)

 * As for superscripts: I rather think that nth is easier to read, the superscript is distracting. What I or you think does not ultimately matter, what matters is that nth looks unprofessional, it is not used by real typographers; any decent manual of style (such as, e.g., the Chicago Manual of Style) will tell you not to use it, and our MoS follows suit.—Emil J. 10:49, 24 August 2010 (UTC)

 * Thanks, Emil. Virginia-American (talk) 12:13, 24 August 2010 (UTC)

Primitive roots of unity modulo n
There are the two articles:
 * Root of unity about complex roots of 1
 * Primitive root modulo n about elements in Z/nZ that generate the multiplicative sub-group of Z/nZ

Now I like to write something about numbers a in Z/nZ with $$a^m = 1$$, where m is chosen to be minimal. I think this is called a primitive root of unity of order m in Z/nZ. The Root of unity article mentions this briefly in the introduction. But the article seems to be so much concerned with complex numbers that I hesitate to add a quite different topic to it. Also adding to Primitive root modulo n may increase confusion about the term primitive root. I tend to start a new article. Any opinions? Suggestions for an article name? HenningThielemann (talk) 22:04, 17 February 2011 (UTC)

I have started such an article with title Root of unity modulo n. HenningThielemann (talk) 18:16, 20 February 2011 (UTC)

 * As you say, this article is about complex roots of unity. There doesn't appear to be an article discussing the concept in other fields, particularly those of characteristic different from 0. (Root of unity modulo n comes relatively close, but isn't the same.) It would be a nice addition if someone would start such a page. I came here looking for a standard definition of a primitive nth root of unity that worked in a field of characteristic p dividing n. I know one can just say it's a generator of the multiplicative group of nth roots of unity in the relevant field (or group of units, I suppose), but that seems to gloss over the inherent separability difference between this case and all others. This page seems cluttered enough that a brief blurb and a reference to a separate article would probably be best. 67.158.43.41 (talk) 16:41, 29 March 2011 (UTC)

Cyclotomic polynomials
This article's section on cyclotomic polynomials is nearly the same length as the article Cyclotomic polynomial. It seems like this material should be very briefly summarized here and merged into that article. (IMO most of it isn't really on topic anyway--for instance, what does it matter to this topic that the first cyclotomic polynomial with a coefficient different from 0, 1, or -1 is the 105th?) 67.158.43.41 (talk) 16:52, 29 March 2011 (UTC)

 * From looking at the history of the "Cyclotomic polynomial" article it looks like for many years it was simply a redirect to this article, so this was the original location for this content. In November 2008, someone created "Cyclotomic polynomial" as its own article and I guess no one removed the content here. I encourage you to prune (rather seriously) the content on cyclotomic polynomials present here! RobHar (talk) 19:05, 29 March 2011 (UTC)

7th roots of unity typo, suggestions
I was trying to verify my own results for the 7th roots of unity, calculating 2*cos(Pi/7) as the solution of the cubic equation,  $$ x^3 + x^2 - 2x - 1 = 0 $$ whose three roots are $$x=2\cos\frac{2n\pi}{7}$$. I found a typo in the posted result online here. The result I get, which appears correct, is $$ \cos(\frac{2\pi,4\pi,6\pi}{7}) = \frac{-1 + \sqrt[3]{\frac{7+21\sqrt{-3}}{2}} + \sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{6}$$ The posted equation is incorrect. The posted equation has a +1 instead of a -1, and a -7 instead of a +7. The author solved $$ x^3 - x^2 - 2x + 1 = 0$$, which gives the solution for $$\cos(\frac{\pi,3\pi,5\pi}{7}) = \frac{1 + \sqrt[3]{\frac{-7+21\sqrt{-3}}{2}} + \sqrt[3]{\frac{-7-21\sqrt{-3}}{2}}}{6}$$, which is incorrect.

I didn't originally intend to calculate or verify the equations for the imaginary portion of the equation $$ i\sin(\frac{2n\pi}{7})$$, but then I couldn't resist finding a more brief equation. $$\sin^2(\frac{2\pi}{7})=\frac{1-\cos(4\pi/7)}{2}$$ which leads to $$\exp(\frac{2\pi i}{7})= \cos(\frac{2\pi}{7}) + \frac{i}{2}\sqrt{2 - 2\cos(\frac{4\pi}{7})}\;\;$$. This is a general equation, applicable to all roots of unity radicals, which usually involve calculating $$2\cos(2\pi/p)$$. By that time, $$2\cos(4\pi/p)$$ is also readily available, so it makes sense that this is probably the best approach to calculating the imaginary part of $$\exp(2\pi i/p)$$. For the case at hand, we are solving a cubic equation. We use s1, and s2, which are complex cuberoots of 1. The s1,s2 terms are to force the cube roots of the complex numbers on the right to be the approriate other cube root, corresponding to the equation on the left, so that the cubic equation result on the right is for $$2\cos(4\pi/7)$$. Assuming my math is correct, this seems like a better formula for the real+imag portion of the desired roots of 7th root unity. Does anyone know the author's of the sourced pdf's email address? I would like to email the author the comments I just posted, if he is interested in updating/correcting his online paper. We need to correct the error in the wikipedia page, even if the author's paper isn't corrected. The link to the author's pdf is http://www.mindspring.com/~jimvb/TheSeventhRootofUnity.pdf Here is my proposed update to the wikipedia page. If nobody has any suggestions or comments, I'll probably update the source page in a few days, once I think there are no further improvements desired.

Given the complex cuberoots of unity, $$s1=e^{2\pi i/3}, s2=e^{4\pi i/3}$$, than one of the seventh roots of unity is $$e^{2\pi i/7}= \frac{-1 + \sqrt[3]{\frac{7+21\sqrt{-3}}{2}} + \sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{6} + \frac{i}{2}\sqrt{\frac{7 - s2\sqrt[3]{\frac{7+21\sqrt{-3}}{2}} - s1\sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{3}}$$

Sheldonison (talk) 19:48, 4 May 2011 (UTC) Sheldon

 * Thanks for noticing this. I've gone ahead and edited the article to reflect your corrections (with a slight notation change) (I also checked your answer myself). I've removed the source as a reference as it's clearly not very reliable. I added a "citation needed" tag for good measure. RobHar (talk) 05:31, 5 May 2011 (UTC)

Cool! I have inspired my very own Wikipedia page update!!! Also thanks for updating the notation, to make it consistent with the rest of the Wikipedia page. I was about to make some similar changes myself, when I noticed you had already updated the Wikipedia page. Again, thanks. Sheldonison (talk) 06:06, 5 May 2011 (UTC) Sheldon

The solution above comes from deriving and solving the cubic equation for $$(x-(\zeta^1+\zeta^{-1}))(x-(\zeta^2+\zeta^{-2}))(x-(\zeta^3+\zeta^{-3}))=0$$. Alternatively, one can derive and solve the cubic equation for $$(x-\zeta^1)(x-\zeta^2)(x-\zeta^{-3})$$, which after solving a quadratic equation for $$(\zeta^1+\zeta^2+\zeta^{-3})$$ leads to the cubic equation $$x^3 + \frac{1 - \sqrt{-7}}{2}x^2 + \frac{-1-\sqrt{-7}}{2}x - 1=0 \;\;$$ one of whose solutions is the following, which although it is completely impossible to tell it, is equal to the other result which was posted. This equivalent result does not have the real and imaginary components of the result separated, although, both involve cube roots of imaginary numbers.

$$\zeta^1=e^{2\pi i/7}=\sqrt[3]{\frac{14-\sqrt{-7}+3\sqrt{21}}{54}} + \omega\sqrt[3]{\frac{14-\sqrt{-7}-3\sqrt{21}}{54}}+\frac{-1+\sqrt{-7}}{6}$$ Sheldonison (talk) 15:57, 6 May 2011 (UTC) Sheldon

I was curious if the imaginary portion could be calculated separately, leading to a more compact result. It apparently can be calculated separately, and the result is slightly simplified, with one less layer of radicals. The key is to first notice that $$2i\sin(2*\pi/7)=\zeta^1-\zeta^{-1}$$. This leads to the following cubic equation $$(x-(\zeta^1-\zeta^{-1}))(x-(\zeta^2-\zeta^{-2}))(x-(\zeta^{-3}-\zeta^{3}))=0$$. One can calculate the sum and products of the three terms, which turns out to be $$\sqrt{-7}$$. Then, after some algebra one gets the following cubic equation: $$x^3 - \sqrt{-7}*x^2 - \sqrt{-7}=0$$ whose roots are 2isin(2*pi/7),2isin(4*pi/7),-2isin(6*pi/7). Solving that cubic, and then doing some arithmetic multiplying by i, gives the following result.

$$2i\sin(2*\pi/7) = \frac{i(\omega^2\sqrt[3]{-13/2\sqrt{7}+3/2\sqrt{-21}} + \omega\sqrt[3]{-13/2\sqrt{7}-3/2\sqrt{-21}} + \sqrt{7})}{6}$$

$$e^{2\pi i/7}= \frac{-1 + \sqrt[3]{\frac{7+21\sqrt{-3}}{2}} + \sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{6} + \frac{i(\omega^2\sqrt[3]{\frac{-13\sqrt{7}+3\sqrt{-21}}{2}} + \omega\sqrt[3]{\frac{-13\sqrt{7}-3\sqrt{-21}}{2}} + \sqrt{7})}{6}$$ Sheldonison (talk) 12:04, 15 May 2011 (UTC)sheldonison

if n is not divisible by the characteristic of the field
My wording about exactly $n$ $n$th roots was a bit reckless, but I did not claim that $z^{n} = 1$ always has distinct roots. BTW I just learned about a (weaker) definition of cyclotomically closed field, which apparently is exactly what we need here. Incnis Mrsi (talk) 18:03, 5 April 2013 (UTC)

Geometry
It would be nice to have a "Geometric interpretation" section that describes the fact that all n roots of unity lie at equidistant points on the unit circle (centered at the origin) in the cartesian/complex plane, i.e., they lie on the vertices of a regular n-gon inscribed within the unit circle. But I don't know where the most appropriate place is to add such a section in the article. — Loadmaster (talk) 19:00, 30 September 2013 (UTC)

 * yes. what is the meaning of this? it seems special 24.98.133.72 (talk) 01:39, 19 August 2014 (UTC)

 * Well, multiplying any combination of the nth roots of 1 always produces one of the (other) roots (or 1) on the complex unit circle. Does that qualify as "special"? There is no deeper meaning, since these are just some of the properties of the unit multiplier (1) in the complex field. — Loadmaster (talk) 19:15, 19 August 2014 (UTC)

Positive power?
Random observation while I was passing through: should the opening paragraph specify that n is positive? After all, raising any complex number (save 0) to the 0th power yields 1. 0 is obviously an integer, but that doesn't mean that the number is a root of unity. (The definition below does a better job, specifying that n must be positive.)

Aasmith (talk) 06:34, 16 September 2015 (UTC)

D.Lazard (talk) 08:14, 16 September 2015 (UTC)

I hate do not understanding stuff!
In this case, what I don't understand is this assertion in the article:
 * Generally, z ∈ R can be considered for any field R

I'm sure this is a stupid question*, but does this mean that in general, a complex number is a real? I guess it depends on the definition of "considered."

Even if not, why talk about complex numbers that are points on the real number line? The imaginary part is zero, so it's a degenerative, trivial complex number.

Right?

For someone who hates not understanding stuff, I sure do it a lot.

__________________
 * I can't stand when people preface questions with that when asking me, but now I see why they do. --Verdana ♥ Bold 07:57, 4 March 2017 (UTC)

 * You are confusing $R$, a variable, which, here, denotes an arbitrary ring, with the standard symbol for the field of real numbers, which is $$\mathbb R$$ or $R$. Nevertheless, I agree that the formulation was confusing, and not only because of the use of $R$ (which was unnecessary). I have thus rewritten the paragraph. D.Lazard (talk) 08:57, 4 March 2017 (UTC)

Roots of Unity Filter
Useful for generating functions http://zacharyabel.com/papers/Multi-GF_A06_MathRefl.pdf Wqwt (talk) 22:32, 1 September 2018 (UTC)
 * This cannot be used in WP, as being original research, not regularly published. Moreover, this result fails to satisfy the WP:notability criteria, because of the lack of WP:secondary sources. D.Lazard (talk) 06:52, 2 September 2018 (UTC)
 * If you consider brilliant.org a valid source then it shows up there too. I'm sure it's in some textbook somewhere, not only for contest math, but probably under a different name. It makes an appearance in Problems from the Book by Titu Andreescu. Wqwt (talk) 02:10, 3 September 2018 (UTC)

Recent edits
Since September, 28, made 25 edits to the articles. A few of them introduce minor improvements, but most do not improve the article and consist mainly of introducing data for roots of unity from 9 to 12. Here are more details on these additions: Thus I'll revert all these edits, and, when useful, I'll add a few words for clarification. Please, do not add data for higher degree, without getting a consensus here (see WP:CONSENSUS and WP:BRD). D.Lazard (talk) 16:20, 29 September 2018 (UTC)
 * Section "Cyclotomic polynomials" is intended for being a summary of the article of the same name (template main article. So adding details for these degrees makes no sense: Either they duplicate the content of the main article or, if they are useful, they should be added to the main article.
 * In section "Algebraic expressions", no explicit formula is given in the edits. There are only general unsourced comments, that either duplicate the beginning of the section, or are not understandable for most readers (references without explanation to casus irreducibilis, or even are completely wrong (I had to revert twice the wrong assertion that eleventh roots of unity cannot expressed in radicals.

In the first illustration ...
... leave off the x- and y-axes!!! Just show the roots of unity —  as the caption states.2600:1700:E1C0:F340:38C4:1D3E:151:63CC (talk) 03:25, 2 October 2018 (UTC)
 * It makes no sense to show the roots of unity without showing where they are in the complex plane. However it may useful to clarify the caption by saying that the roots are the blue dots. This has been done. D.Lazard (talk) 08:35, 2 October 2018 (UTC)

title: Talk:Root of unity/Archive 1


text:
Jiang

Jiang may refer to:

 * Jiang (rank), rank held by general officers in the military of China
 * Jiang (surname), several Chinese surnames
 * Jiang Zemin (1926–2022), as general secretary of the Chinese Communist Party
 * Jiang River, an ancient river of China
 * Jiang County, in Shanxi, China
 * Fermented bean paste, known as 酱 (jiàng) in Chinese

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Jiang


text:
Theoreticism

In philosophy and particularly political philosophy, theoreticism is the preference for theory over practice (or, more broadly, abstract knowledge over concrete action), or a philosophical position which would lead to such a preference.

The term is often used pejoratively. In Marxist philosophy, for instance, theoreticism is often identified as a political error, valorizing the efforts of academic Marxism over those of revolutionary struggle. Louis Althusser, for instance, criticized his own early work for theoreticism. In phenomenology, theoreticism would be something closer to the over-valuing of knowledge at the cost of losing a proper appreciation of experience. Martin Heidegger claimed this trend was begun by Plato, and that it continued in an "intensification and hardening of 'theoreticism', the drive toward technical and objectifying modes of knowledge and, with it, the oblivion of any more primordial or more reverential kind of existence."

title: Theoreticism


text:
Archigram

Archigram was an avant-garde British architectural group whose unbuilt projects and media-savvy provocations "spawned the most influential architectural movement of the 1960's," according to Peter Cook, in the Princeton Architectural Press study Archigram (1999). Neofuturistic, anti-heroic, and pro-consumerist, the group drew inspiration from technology in order to create a new reality that was expressed through hypothetical projects, i.e., its buildings were never built, although the group did produce what the architectural historian Charles Jencks called "a series of monumental objects (one hesitates in calling them buildings since most of them moved, grew, flew, walked, burrowed or just sank under the water." The works of Archigram had a neofuturistic slant, influenced by Antonio Sant'Elia's works. Buckminster Fuller and Yona Friedman were also important sources of inspiration.

"Their attitude was closely tied to the technocratic ideology of the American designer Buckminster Fuller," Kenneth Frampton confirms, in Modern Architecture: A Critical History, "and to that of his British apologists John McHale and Reyner Banham. ... Archigram's subsequent commitment to a 'high-tech,' lightweight, infrastructural approach (the kind of indeterminacy implicit in the work of Fuller and even more evident in Yona Friedman's L'Architecture mobile of 1958) brought them, rather paradoxically, to indulge in ironic forms of science fiction, rather than to project solutions that were either truly indeterminate or capable of being realized and appropriated by society."

Origins: 1960-61
Based at the Architectural Association in London, the main members of the group were Peter Cook, Warren Chalk, Ron Herron, Dennis Crompton, Michael Webb and David Greene. Archigram formed in late in the year 1960, shortly before the first issue of their magazine of the same name, which appeared in 1961. Designer Theo Crosby was the "hidden hand" behind the group. He gave them coverage in Architectural Design magazine (where he was an editor from 1953 to 1962), brought them to the attention of the Institute of Contemporary Arts (ICA) in London, where, in 1963, they mounted an exhibition called Living City, and in 1964 brought them into the Taylor Woodrow Design Group, which he headed, to take on experimental projects. The pamphlet Archigram I was printed in 1961 to proclaim their ideas. The group experimented with modular technology, mobility through the environment, space capsules, and consumer-culture imagery. Their projects offered a seductive vision of a glamorous, high-tech future. Social and environmental issues were, however, left largely unaddressed.

The group tapped into the zeitgeist captured by Richard Hamilton in his "This Is Tomorrow" exhibition in 1956 at the Whitechapel Art Gallery, by Pop Art, by the turned-on, tuned-in psychedelic counterculture, and by the gnomic pronouncements of the media theorist Marshall McLuhan. Then, too, some of its guiding principles were premonitory of what the politically radical French avant-garde would later call Situationism. In the Living City exhibition, Archigram "collected images from any part of the city—the accepted Pop iconography of spaceman, superman, robotman, and woman—but presented them in a way and with a message that was new to architecture," Jencks writes, in Modern Movements in Architecture.

"The city was seen not as architecture (hardware), but as people and their 'situations' (software). It was these infinitely variable and fleeting situations which gave the real life to the city: in this sense 'the home, the whole city, and the frozen pea pack are all the same.' Not only are they all expendable, but they are all products which interact with man in the same level, the situation." Archigram agitated to prevent modernism from becoming a sterile orthodoxy, rendered safe by its adherents. Contrary to Buckminster Fuller's notion of "ephemeralization," which assumes more must be done with less and less (because material resources are finite), Archigram presumes a future of inexhaustible resources.

Mid-'60s
According to Jencks, Archigram's "extraordinary inventiveness" and delirious, Pop sci-fi imagery attracted international media attention throughout 1963–65. The group designed cities "that looked like computers and molehills, that crawled on the shoots of a telescope like Eduardo Paolozzi's Bug-Eyed Monsters, that bobbed under the sea like so many skewered balloons, that sprouted—swock!—out of the sea like a Tom Wolfian, hydraulic umbrella, that zoomed down from the clouds flashing 'Destroy-Man! Kill-All-Humans,' a space-comic-robot-zaap, that clicked into place along pneumatic tubes, a plug-in plastic layer cake, that gurgled and spluttered over the old city like creeping, cancerous, testubular, friendly Daleks."

"The strength of Archigram's appeal," wrote the architecture critic and historian Reyner Banham, "stems from many things, including youthful enthusiasm in a field (city planning) which is increasingly the preserve of middle-aged caution. But chiefly it offers an image-starved world a new vision of the city of the future, a city of components on racks, components in stacks, components plugged into networks and grids, a city of components being swung into place by cranes."

Late '60s
By 1967, in works like "Control and Choice Living" (1967), the group had turned its attention to the question of exploiting, in architecture and urban planning, those "systems, organizations, and techniques that permit the emancipation and general good life of the individual" within "a high-density location," writes Jencks. "The solution was a minimal set of fixed elements which increased in flexibility from the permanent pylons to the completely flexible 'air-habs.' The latter invention was a combination un-house and blow-up satellite (that is, an air-inflatable satellite)" of seemingly infinite possibility. The inhabitant "could dial out a room or if this were not desired drive the electric car into it and sprout out a room within a room. In effect, the services robot is now decentralized to include every part of the house."

1970's
By the early 1970s, the group had changed its strategy. In 1973, wrote Theo Crosby, its members had "found their original impulses towards megastructures blunted by the changing intellectual climate in England, where the brash dreams of modern architects are received with ever-increasing horror. They are now more concerned with the infiltration of technology into the environment at a much less obvious level."

If we consider for a moment Christo's seminal work – the 'wrapped cliff' – we might see it in one of two ways: as a wrapped cliff or, preferably, as the point at which all other cliffs are unwrapped. An Archigram project attempts to achieve this same altered reading of the familiar (in the tradition of Buckminster Fuller's question, 'How much does your building weigh?'). It provides a new agenda where nomadism is the dominant social force; where time, exchange, and metamorphosis replace stasis; where consumption, lifestyle, and transience become the programme; and where the public realm is an electronic surface enclosing the globe —David Greene

The group was supported by mainstream architects, such as David Rock of BDP. Rock later nominated Archigram for the RIBA Royal Gold Medal, which they received in 2002.

Plug-in-City, Peter Cook, 1966
Plug-in-City is a mega-structure with no buildings, just a massive framework into which dwellings in the form of cells or standardised components could be slotted. The machine had taken over and people were the raw material being processed, the difference being that people are meant to enjoy the experience.

The Walking City, Ron Herron, 1964
The Walking City is constituted by intelligent buildings or robots in the form of giant, self-contained living pods designed to roam freely. The form derived from a combination of insect and machine and was a literal interpretation of Le Corbusier's aphorism that a house was a "machine for living in." The pods were intended to be independent yet parasitic, since they could "plug into" way stations to exchange occupants or replenish resources. The imagined context for these ambulatory, high-tech structures was a post-apocalyptic future in which the urban landscape lay in ruins, devastated by a man-made catastrophe.

Archigram was interested "in the Armageddon overtones of survival technology," Frampton claims. "For all their surface irony, [Herron's "Walking Cities"] were clearly projected as stalking across a ruined world in the aftermath of a nuclear war. ... [T]hey suggest some sort of nightmarish salvation, rescuing both men and artifacts after a cataclysmic disaster."

Instant City, Peter Cook, 1968-70
Instant City is a mobile technological event that drifts into underdeveloped, drab towns via air (balloons) with provisional structures (performance spaces) in tow. The effect is a deliberate overstimulation to produce mass culture, with an embrace of advertising aesthetics. The whole endeavor is intended to eventually move on, leaving behind advanced technology hook-ups.

Other projects
Tuned City, in which Archigram's infrastructural and spatial additions attach themselves to an existing town at a percentage that leaves evidence of the previous development, rather than subsuming the whole.

The French fashion line Sixpack France dedicated their Summer Spring 2009 Collection to this movement.

Influence and legacy
The group's paper architecture (i.e., visionary architecture) served as a source of inspiration for early works by Norman Foster, Gianfranco Franchini, and Future Systems and, most memorably, the Pompidou centre (commissioned 1971, opened 1977) by Richard Rogers and Renzo Piano, an arresting example of High tech, a.k.a. structural expressionism. "The building is obviously a realization of the technological and infrastructural rhetoric of Archigram," writes Frampton, "and while the full consequences of this approach are becoming evident through everyday use, it is apparent that certain paradoxical achievements may be claimed on its behalf."

"In the first place, it is an outstanding popular success—as much for its sensational nature as anything else. In the second, it is a brilliant tour de force in advanced technique, looking for all the world like an oil refinery whose technology it attempts to emulate."

Frampton concedes, however, that the Pompidou seems "to have come into being with the minimum regard for the specificity of its brief—for the art and library holdings it was destined to house. It represents the design approach of indeterminacy and optimum flexibility taken to extremes."

In Jencks's estimation, "the great contribution of the British avant-garde"—of which Archigram is perhaps the most exuberantly iconoclastic exponent, in architecture—"has been to open up and develop new attitudes towards living in an advanced industrial civilization where only stereotyped rejection had existed before, to dramatizing consumer choice and communicating the pleasure inherent in manipulating sophisticated technology. If these strategies will not solve the deeper social and political urban problems, at least they open up new alternative routes for thinking about consumer society and urbanism."

There were detractors. "By 1972, Robert Venturi and Denise Scott Brown could no longer take Archigram seriously," writes Simon Sadler, in Archigram: Architecture without Architecture. He quotes their landmark critique of postmodern architecture, Learning from Las Vegas, published that year: “Archigram’s structural visions are Jules Verne versions of the Industrial Revolution with an appliqué of Pop-aerospace terminology.”

And defenders: "Three years later," writes Sadler, the architecture critic Martin Pawley argued that Archigram "stood for 'an existential technology for individuals that the world will, in time, come to regard with the same awe as is presently accorded to the prescience of Jules Verne, H. G. Wells, or the Marquis de Sade. Futile to complain (as many do),‘But they never build anything.’ Verne never built the Nautilus, Wells could hardly drive a car, and the Marquis de Sade?”

In 2019, the M+ museum in Hong Kong acquired Archigram's entire archive, despite purported attempts to block the sale to an overseas buyer.

title: Archigram


text:
Caribbean Basin Trade and Partnership Act

The Caribbean Basin Trade Partnership Act (CBTPA) is a law adopted by the U.S. Government in October 2000 to delineate enhanced trade preferences and eligibility requirements for the 24 beneficiary countries of the Caribbean Basin region.

On October 2, President Clinton signed the Proclamation implementing the Caribbean Basin Trade Partnership Act (CBTPA – Title II of the Trade and Development Act of 2000). The Presidential Proclamation declares the 24 current beneficiary countries of the Caribbean Basin Initiative (CBI) to be "Beneficiary Countries" for purposes of the enhanced trade preferences made available under the CBTPA. In addition, the Proclamation modifies the Harmonized Tariff Schedule to reflect the new trade preferences. It also delegates to the Office of the United States Trade Representative the authority to publish (through a Federal Register notice) additional determinations regarding the compliance of CBTPA Beneficiary Countries with customs-related procedures established in the CBTPA.

Summary of Enhanced CBTPA Preferences
The CBTPA significantly expands preferential treatment for apparel made in the Caribbean Basin region. Duty- and quota-free treatment is provided for apparel made in the CBI from U.S. fabrics formed from U.S. yarns. Duty/quota-free treatment is also available for certain knit apparel made in CBTPA beneficiary countries from fabrics formed in the Caribbean Basin region, provided that U.S. yarns are used in forming the fabric. This "regional fabric" benefit for knit apparel is subject to an overall yearly limit, with a separate limit provided for T-shirts.

New duty/quota free treatment will also be available for apparel made in the CBI from fabrics determined to be in "short supply" in the United States, and for designated "hand-loomed, handmade, or folklore" articles.

In addition to these apparel preferences, the CBTPA provides NAFTA-equivalent tariff treatment for certain items previously excluded from duty-free treatment under the CBI program (e.g., footwear, canned tuna, petroleum products, watches and watch parts).

Beneficiary Country Designation
The CBTPA authorized the President to designate individual countries as being "Beneficiary Countries" in order to receive the enhanced trade benefits available under the Act. The twenty-four current beneficiaries of the Caribbean Basin Economic Recovery Act (CBERA) were potentially eligible to be declared CBTPA Beneficiary Countries. These countries are: Antigua and Barbuda, Aruba, Bahamas, Barbados, Belize, Costa Rica, Dominica, Dominican Republic, El Salvador, Grenada, Guatemala, Guyana, Haiti, Honduras, Jamaica, Montserrat, Netherlands Antilles, Nicaragua, Panama, Saint Kitts and Nevis, Saint Lucia, Saint Vincent and the Grenadines, Trinidad and Tobago, and British Virgin Islands. Through the Proclamation issued today, President Clinton designated all 24 countries as CBTPA Beneficiary Countries.

Eligibility Review Factors
In considering the eligibility of these countries for CBTPA Beneficiary Country status, the CBTPA required the President to take into account the existing eligibility criteria of the CBERA, as well as several new criteria elaborated in the CBTPA. These new criteria include:

 * 1) Whether the beneficiary country has demonstrated a commitment to undertake its obligations under the WTO on or ahead of schedule and participate in negotiations toward the completion of the FTAA or another free trade agreement.
 * 2) The extent to which the country provides protection of intellectual property rights consistent with or greater than the protection afforded under the Agreement on Trade-Related Aspects of Intellectual Property Rights.
 * 3) The extent to which the country provides internationally recognized worker's rights, including:
 * 4) the right of association;
 * 5) the right to organize and bargain collectively;
 * 6) a prohibition on the use of any form of forced or compulsory labor;
 * 7) a minimum age for the employment of children; and
 * 8) acceptable conditions of work with respect to minimum wages, hours of work, and occupational safety and health;
 * 9) Whether the country has implemented its commitments to eliminate the worst forms of child labor.
 * 10) The extent to which the country has met U.S. counter-narcotics certification criteria under the Foreign Assistance Act of 1961.
 * 11) The extent to which the country has taken steps to become a party to and implements the Inter-American Convention Against Corruption.
 * 12) The extent to which the country applied transparent, nondiscriminatory and competitive procedures in government procurement, and contribute to efforts in international for to develop an implement rules on transparency in government procurement.

Customs-Related Eligibility Determination
In addition to Presidential eligibility designations based on the criteria described above, the CBTPA requires an additional determination that countries have implemented or are making substantial progress towards implementing certain customs procedures based on those contained in the NAFTA. Today's Proclamation delegates authority for these additional determinations to USTR. USTR will publish initial determinations on this eligibility criteria in coming days, through a notice in the Federal Register.

Eligibility Review Process
The USTR-chaired Trade Policy Staff Committee (TPSC) conducted a review of countries' eligibility for CBTPA preferences, taking into account the criteria established in the Act. This review relied upon information provided by U.S. Embassies, keyed to the various eligibility criteria, as well as on information from other reliable sources, such as the International Labour Organization. In addition, the TPSC took into account public comments regarding the eligibility review, solicited through a Federal Register notice published June 19, 2000. The TPSC received 206 comments in response to this notice.

Through a series of meetings, the TPSC identified a number of concerns regarding certain countries' performance with respect to the eligibility criteria established in the CBTPA. For each of these areas of concern, the TPSC further identified specific policy objectives to be pursued with the relevant governments, with the general aim of soliciting assurances that these concerns would be addressed. For those countries for which concerns and objectives had been identified, U.S. Embassy officials in the Caribbean Basin region and other U.S. government officials conducted intensive advocacy with local government officials. TPSC recommendations to the President regarding the designation of CBTPA Beneficiary Countries were based on the results of these efforts.

The eligibility review process involved direct, issue-specific advocacy with a majority of the countries that were potentially eligible for CBTPA benefits. Objectives with respect to intellectual property protection, worker rights, implementation of WTO agreements, and commitments to abide by international anti-corruption guidelines were pursued with a range of countries. The review concluded that CBI countries satisfied the CBTPA criteria regarding commitments to eliminate the worst forms of child labor; however, several countries were urged to expand upon their current efforts to combat all forms of child labor.

Guatemala: Additional Review of Worker Rights Issues
The review of Guatemala's eligibility for the enhanced CBI preferences involved extensive consideration of the worker rights situation in that country. The United States raised specific concerns with respect to anti-union violence, including a 1999 incident in which armed vigilantes threatened and kidnapped leaders of a banana workers' union who were protesting the illegal dismissal of 900 workers. Guatemala was asked to facilitate negotiations to reemploy the 900 fired workers and to commit to speedy and effective implementation of labor code requirements with respect to this case. Guatemala's government was also asked to work with the country's legislature to pass proposed revisions to the labor code, and to provide a commitment, over the longer-term, to continual improvements in law enforcement and judicial administration related to the protection of worker rights.

In their responses to U.S. concerns, Guatemala's Vice President, Minister of Labor, and other officials have demonstrated considerable good faith in seeking to improve the worker rights situation in their country. The United States welcomes the Ministry of Labor's efforts to date to facilitate a resolution to the situation arising from the 1999 incident involving banana workers. There is evidence of progress in prosecuting those responsible for violence against workers in that case. We also welcome efforts by the government to suspend operating licenses of companies which have violated labor code provisions. In addition, the Guatemalan executive branch has presented legislation to bring the country's labor laws into conformity with ILO recommendations.

It is on the basis of these actions and assurances that Guatemala has been designated at this time as a CBTPA Beneficiary Country.

Despite certain forthcoming actions and statements by Guatemalan officials, the United States remains deeply concerned that the overall worker rights environment in Guatemala represents a threat to those seeking to advance basic, internationally recognized rights for workers. Instances of anti-union violence, including occasional murders, persist. The widespread impunity for those who provoke and carry out such violence is a particularly severe concern.

Consequently, Guatemala's CBTPA beneficiary status will be reviewed in April 2001, with a focus on further improvements in the area of worker rights. This review will include the following objectives: a) an assessment that the Guatemalan executive branch is taking all actions within its authority to ensure the physical safety and human and civil rights of union leaders and the effective criminal prosecution of persons charged with provoking anti-union violence, including killings of union leaders; b) assurances that the Government of Guatemala is taking all steps within its power to provide for the re-employment of the 900 fired banana workers and settlement of related labor law violations; c) further progress towards enacting the new Labor Code; and d) further improvements in labor law enforcement and judicial administration related to the protection of labor rights.

As a further sign of the seriousness with which the United States views these issues, Ambassador Barshefsky is initiating immediately a review of Guatemala's eligibility as a beneficiary country under the Generalized System of Preferences (GSP). This unprecedented self-initiated review will also be concluded in April 2001, and will focus on the government's response to anti-union violence and other aspects of internationally recognized worker rights.

Worker Rights Monitoring in El Salvador, Honduras, and Nicaragua
Worker rights issues were also pursued in the context of the CBTPA eligibility review of El Salvador, Honduras, and Nicaragua. With respect to El Salvador, the United States raised concerns regarding the effect of certain privatization programs in restricting union activity, as well as excessive legal formalities applied to the establishment of trade unions. In Honduras, the United States focused on the government's efforts to revise the Honduran labor code to reflect recommendations made by the International Labour Organization. In Nicaragua, the U.S. expressed particular concern with respect to anti-union activity at two apparel factories in the Las Mercedes Free Trade Zone, and successfully sought the government's assurances that workers at those factories would be informed of their rights under Nicaragua's Labor Code.

In each of these cases, the governments in question provided responses which were helpful in addressing U.S. concerns. Nonetheless, the Administration believes that worker rights practices in these countries should be subject to ongoing monitoring. This monitoring will focus on follow-through to the commitments made by these governments in the context of the CBTPA eligibility review. In addition, the United States will request bilateral consultations with each of the three governments to discuss worker rights concerns by June 30, 2001.

''Material taken from http://www.usleap.org/trade/CTPA10-03.html as well as


text:
Paper candidate

In a representative democracy, a paper candidate (also known as a no-hope candidate) is a candidate who stands for a political party in an electoral division where the party in question enjoys only low levels of support. Although the candidate has little chance of winning, a major party will normally make an effort to ensure it has its name on the ballot paper in every constituency. In two-party systems, a paper candidate may also be known as token opposition. In Dutch-speaking countries, the last candidate on a party list is called a lijstduwer and are usually well-known non-politicians such as artists, celebrities and sports people to attract more votes for the party.

A dummy candidate is similar to a paper candidate in that both types do not expect to win their race; however, they differ in that a dummy candidate typically has an ulterior motive for being in the race, such as to dishonestly divert votes away from more legitimate candidates or to take advantage of benefits afforded to political candidates. Another related concept is the stalking horse candidate, a junior or little-known politician who challenges an incumbent to test the support for a more serious challenge from someone else.

United States
Paper candidates may be local party members, or members from neighbouring areas or from central office depending on the laws of the jurisdiction; most U.S. states, for example, require all candidates for state and local office to reside in the district they intend on representing for a certain length of time. The main purpose of fielding paper candidates is to maintain or improve the profile of a political party and, in two-party systems, to provide at least nominal opposition to a seemingly unassailable incumbent. Another potential use for paper candidacy is to allow a candidate who wants off the ballot of another race to do so by running for something else, a race they cannot possibly win (such as Rick Lazio, who lost a Republican primary for New York Governor in 2010 but still had a third-party ballot line; in order to disqualify himself from the gubernatorial election Lazio was nominated for a judicial seat in the Bronx that was so heavily Democratic that he could not have possibly won if he wanted to, and he did not). The paper candidates themselves typically do little or no campaigning and neither incur nor claim any expenses.

There are circumstances where a paper candidate can win an election, often when the opposing candidate is unexpectedly embroiled in scandal; for example, then-27-year-old American Chris Smith, who ran as a token opponent to New Jersey congressman Frank Thompson in 1978 and 1980, won the latter contest after Thompson was embroiled in the Abscam scandal. Another example is Michael Patrick Flanagan, a little-known Republican attorney who defeated longtime Democratic incumbent Dan Rostenkowski, the onetime powerful Chair of the House Ways and Means Committee, in a heavily Democratic Chicago-based district after Rostenkowski was indicted on charges of mail fraud. The victory occurred in 1994 as part of a national Republican wave.

In jurisdictions with strict campaign finance and spending laws governing elections, it is often a legal necessity to run the maximum number candidates permitted (or close thereto) in order to spend the maximum amount of money otherwise permitted by law. In such cases, parties will run paper candidates, but will usually endeavour to use the extra spending allowance afforded (at least to the extent that it may be permitted in the jurisdiction) to campaign in districts where they have a realistic chance of winning.

United Kingdom
In the United Kingdom, major parties often find it difficult to field a full list of candidates for all council seats up for election, especially in the case of councils with "all-up" elections. Parties find it desirable to persuade people to stand as paper candidates so that:

 * Supporters have an opportunity to vote for the party
 * The total vote obtained across the council and the nation is maximised
 * All seats are contested so there is no risk that candidates from other parties can be declared elected unopposed

In Britain, being nominated as a local election candidate simply involves signing some forms, with no deposit required. A paper candidate will often do no campaigning at all and so be able to submit a zero return of election expenses, simplifying the paperwork for the election agent. Paper candidates are commonly fielded in different locations by all the major parties in both local and national elections.

Some paper candidates stand in order to help their party but do not wish to be elected to the post in question. In fact, some paper candidates only agree to stand after receiving assurances that there is no "risk" of them getting elected. Following the Scottish National Party's landslide in Scotland in the 2015 general election, when they went from 6 to 56 seats (out of 59 Scottish seats), it was reported that some candidates were so surprised at winning they considered resigning immediately after the election. In the 2019 general election, some Conservative candidates were selected just weeks prior to the election and won seats in the Labour "red wall" in northern England and Wales, which were previously considered unwinnable by the party.

In Belfast North at the 2017 election, the Social Democratic and Labour Party (SDLP) were criticised for standing Martin McAuley as a "paper candidate" — a deliberately weak candidate, a 26-year-old with little experience, on the assumption that nationalist and Catholic voters would instead vote for Sinn Féin, instead of the unionist candidate Nigel Dodds (DUP). McAuley rejected the label of "paper candidate" Dodds narrowly won the election. In 2019, the SDLP did not field a candidate in Belfast North and Sinn Féin won the seat. The SDLP were also accused of standing a "paper candidate," Mary Garrity, in Fermanagh and South Tyrone in 2017, in order to help Sinn Féin win the seat from the UUP.

Ireland
In Ireland, the single transferable vote system is used, so paper candidates are less common and more seats are up for grabs. However, there have been cases of unexpected surges for certain parties (such as Labour in 1992 and Sinn Féin in 2020) that led to many paper candidates being unexpectedly elected. This was epitomised in 1992 with Moosajee Bhamjee, Ireland's first Indian and first Muslim TD, who only stood in Clare because no one else in the local Labour party wanted to. In 2020, many inexperienced Sinn Féin candidates won, such as Claire Kerrane (Roscommon–Galway) and Violet-Anne Wynne (Clare), both of whom had never held an elected office before and were the first Sinn Féin TDs in those counties since the 1920s. On the other hand, the success of the Social Democrats in the 2020 election was attributed to their lack of paper candidates: they fielded candidates in just 20 out of 39 constituencies and won 6 TDs.

Canada
In Canada, paper candidates may exist at both the federal and provincial / territorial levels, while the term does not apply in municipal elections as official parties do not exist at the city council level. As in Great Britain, they most commonly exist to allow the main political parties to field candidates in as many constituencies as possible. From 2004 to 2011, parties at the federal level had a financial incentive to draw as many votes as possible due to a per-vote subsidy, paid to all parties with at least 2% of the total popular vote regardless of the number of seats actually won. This payment, enacted under a Liberal government, was fully eliminated in 2015 under a Conservative government.

Nevertheless, even without per-vote subsidies, parties have other strong incentives to maximize their overall popular vote. In particular, even though winning the popular vote in itself conveys no special rights or privileges in the Canadian electoral system, there have been multiple instances (especially at the provincial level) where a party that has won the popular vote but not a plurality of seats has attempted to claim a mandate to govern, often using its popular vote victory to justify such claims. Political parties therefore have an incentive to ensure at all costs that the entire electorate can at least vote for them.

Paper candidates are sometimes used if the party is not seriously contesting the election but must run candidates so it can either get registered or stay registered for some other purpose. In such cases, paper candidates will usually run in districts where ideologically like-minded parties are seen to have little chance of winning to minimize the risk of any nominal support they might receive proving to be the decisive margin in a close local election under Canada's first-past-the-post electoral system.

An example of this scenario in action was found in Saskatchewan, where the Progressive Conservative Party withdrew from public presence in the late 1990s but continued to run at least ten candidates, all in urban ridings where the then-governing Saskatchewan New Democratic Party was dominant at the time, in the province's general elections until the relevant law was amended so as to keep its registration with Elections Saskatchewan, largely to avoid losing control of what at the time was believed to be a substantial amount of money. (Most of the PCs' former politicians and grassroots members formed or joined the Saskatchewan Party.) The Progressive Conservative Party has since attempted to "revive" itself and no longer considers its nominees to be paper candidates. However, they have only achieved modest levels of support with candidates affiliated with the national Conservative Party typically running under the banner of the Progressive Conservative Party in provincial elections.

In neighbouring Alberta, the United Conservative Party formed from a "merger" of the Progressive Conservative and Wildrose parties. However, as Albertan electoral law did not permit the parties to merge officially, both parties continue to exist on paper under the UCP leadership team. Albertan electoral law only requires a party to nominate one candidate to maintain its registration, so in the 2019 provincial election, each party ran one candidate in Edmonton-Strathcona, the riding represented by the incumbent New Democratic premier, Rachel Notley, which is widely seen to be the NDP's safest seat. The PC candidate polled enough votes to finish in fourth place, ahead of candidates for parties running "serious" campaigns such as the Alberta Liberal Party.

Paper candidates in Quebec ("poteau" in Canadian French, suggesting that the candidate will only be seen on utility poles), particularly of the name-on-ballot variety, can sometimes provide unwanted attention for the candidate's party, particularly if they become viable prospects for elections. For example, in the 2011 federal election, a sudden increase in opinion-poll support, particularly in Quebec, for the New Democratic Party, which historically had a minimal presence in that province, led to greater scrutiny of some of that party's lower-profile Quebec candidates. One such candidate, Ruth Ellen Brosseau, was working as a bartender hundreds of kilometres away from the riding of Berthier—Maskinongé where she was on the ballot. She won even though she had never been in her riding, claimed no expenses for her campaign, and spoke its dominant language (Canadian French) poorly. However, Brosseau successfully shook off the label by the time of the 2015 election, having become recognized as a hard-working MP who had built a significant base of popularity in her district. She was narrowly defeated in the 2019 and 2021 elections.

The federal Progressive Conservative Party also had several paper candidates who won election in the party's historic landslide victory in the 1984 election, such as Thomas Suluk in Nunatsiaq.

An extreme version of a paper candidate is a "Name on Ballot", often referred to by the acronym "NoB". Many NoBs (Names on Ballot) will only place campaign signs, and some do not even campaign at all. In most cases, the only requirement is that the candidate show up at the returning officer's headquarters for a few moments to take an oath and pay the required nomination deposit. In a more extreme example, in Alberta, candidates do not need to show up to talk to a returning officer, as long as someone on behalf of the party drops off the required paperwork and funds. Many smaller parties, such as the Prince Edward Island New Democrats, will field most of their candidates in any given election as NoBs. The term is often worn as a badge of pride in one's loyalty to the party. PEI New Democrat, Dr. Bob Perry, who has been a NoB many times in the past, often calls himself "Dr. NoB" at election time.

title: Paper candidate