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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod $p$ when $p$ is a prime number.

The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number $p$ and every positive integer $k$ there are fields of order $p^{k}$, all of which are isomorphic.

Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.

Properties
A finite field is a finite set that is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms.

The number of elements of a finite field is called its order or, sometimes, its size. A finite field of order $q$ exists if and only if $q$ is a prime power $p^{k}$ (where $p$ is a prime number and $k$ is a positive integer). In a field of order $p^{k}$, adding $p$ copies of any element always results in zero; that is, the characteristic of the field is $p$.

If $q = p^{k}$, all fields of order $q$ are isomorphic (see  below). Moreover, a field cannot contain two different finite subfields with the same order. One may therefore identify all finite fields with the same order, and they are unambiguously denoted $$\mathbb{F}_{q}$$, $F_{q}$ or $GF(q)$, where the letters GF stand for "Galois field".

In a finite field of order $q$, the polynomial $X^{q} − X$ has all $q$ elements of the finite field as roots. The non-zero elements of a finite field form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. (In general there will be several primitive elements for a given field.)

The simplest examples of finite fields are the fields of prime order: for each prime number $p$, the prime field of order $p$ may be constructed as the integers modulo $p$, $$\mathbb{Z}/p\mathbb{Z}$$.

The elements of the prime field of order $p$ may be represented by integers in the range $0, ..., p − 1$. The sum, the difference and the product are the remainder of the division by $p$ of the result of the corresponding integer operation. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see ).

Let $F$ be a finite field. For any element $x$ in $F$ and any integer $n$, denote by $n ⋅ x$ the sum of $n$ copies of $x$. The least positive $n$ such that $n ⋅ 1 = 0$ is the characteristic $p$ of the field. This allows defining a multiplication $(k, x) ↦ k ⋅ x$ of an element $k$ of $GF(p)$ by an element $x$ of $F$ by choosing an integer representative for $k$. This multiplication makes $F$ into a $GF(p)$-vector space. It follows that the number of elements of $F$ is $p^{n}$ for some integer $n$.

The identity

(sometimes called the freshman's dream) is true in a field of characteristic $p$. This follows from the binomial theorem, as each binomial coefficient of the expansion of $(x + y)^{p}$, except the first and the last, is a multiple of $p$.

By Fermat's little theorem, if $p$ is a prime number and $x$ is in the field $GF(p)$ then $x^{p} = x$. This implies the equality $$X^p-X=\prod_{a\in \mathrm{GF}(p)} (X-a)$$ for polynomials over $GF(p)$. More generally, every element in $GF(p^{n})$ satisfies the polynomial equation $x^{p^{n}} − x = 0|undefined$.

Any finite field extension of a finite field is separable and simple. That is, if $E$ is a finite field and $F$ is a subfield of $E$, then $E$ is obtained from $F$ by adjoining a single element whose minimal polynomial is separable. To use a piece of jargon, finite fields are perfect.

A more general algebraic structure that satisfies all the other axioms of a field, but whose multiplication is not required to be commutative, is called a division ring (or sometimes skew field). By Wedderburn's little theorem, any finite division ring is commutative, and hence is a finite field.

Existence and uniqueness
Let $q = p^{n}$ be a prime power, and $F$ be the splitting field of the polynomial $$P = X^q-X$$ over the prime field $GF(p)$. This means that $F$ is a finite field of lowest order, in which $P$ has $q$ distinct roots (the formal derivative of $P$ is $P = −1$, implying that $gcd(P, P) = 1$, which in general implies that the splitting field is a separable extension of the original). The above identity shows that the sum and the product of two roots of $P$ are roots of $P$, as well as the multiplicative inverse of a root of $P$. In other words, the roots of $P$ form a field of order $q$, which is equal to $F$ by the minimality of the splitting field.

The uniqueness up to isomorphism of splitting fields implies thus that all fields of order $q$ are isomorphic. Also, if a field $F$ has a field of order $q = p^{k}$ as a subfield, its elements are the $q$ roots of $X^{q} − X$, and $F$ cannot contain another subfield of order $q$.

In summary, we have the following classification theorem first proved in 1893 by E. H. Moore:

''The order of a finite field is a prime power. For every prime power $q$ there are fields of order $q$, and they are all isomorphic. In these fields, every element satisfies'' $$x^q=x,$$ and the polynomial $X^{q} − X$ factors as $$X^q-X= \prod_{a\in F} (X-a).$$

It follows that $GF(p^{n})$ contains a subfield isomorphic to $GF(p^{m})$ if and only if $m$ is a divisor of $n$; in that case, this subfield is unique. In fact, the polynomial $X^{p^{m}} − X|undefined$ divides $X^{p^{n}} − X|undefined$ if and only if $m$ is a divisor of $n$.

Non-prime fields
Given a prime power $q = p^{n}$ with $p$ prime and $n > 1$, the field $GF(q)$ may be explicitly constructed in the following way. One first chooses an irreducible polynomial $P$ in $GF(p)[X]$ of degree $n$ (such an irreducible polynomial always exists). Then the quotient ring $$\mathrm{GF}(q) = \mathrm{GF}(p)[X]/(P)$$ of the polynomial ring $GF(p)[X]$ by the ideal generated by $P$ is a field of order $q$.

More explicitly, the elements of $GF(q)$ are the polynomials over $GF(p)$ whose degree is strictly less than $n$. The addition and the subtraction are those of polynomials over $GF(p)$. The product of two elements is the remainder of the Euclidean division by $P$ of the product in $GF(p)[X]$. The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see .

However, with this representation, elements of $GF(q)$ may be difficult to distinguish from the corresponding polynomials. Therefore, it is common to give a name, commonly $α$ to the element of $GF(q)$ that corresponds to the polynomial $X$. So, the elements of $GF(q)$ become polynomials in $α$, where $P(α) = 0$, and, when one encounters a polynomial in $α$ of degree greater or equal to $n$ (for example after a multiplication), one knows that one has to use the relation $P(α) = 0$ to reduce its degree (it is what Euclidean division is doing).

Except in the construction of $GF(4)$, there are several possible choices for $P$, which produce isomorphic results. To simplify the Euclidean division, one commonly chooses for $P$ a polynomial of the form $$X^n + aX + b,$$ which make the needed Euclidean divisions very efficient. However, for some fields, typically in characteristic $2$, irreducible polynomials of the form $X^{n} + aX + b$ may not exist. In characteristic $2$, if the polynomial $X^{n} + X + 1$ is reducible, it is recommended to choose $X^{n} + X^{k} + 1$ with the lowest possible $k$ that makes the polynomial irreducible. If all these trinomials are reducible, one chooses "pentanomials" $X^{n} + X^{a} + X^{b} + X^{c} +  1$, as polynomials of degree greater than $1$, with an even number of terms, are never irreducible in characteristic $2$, having $1$ as a root.

A possible choice for such a polynomial is given by Conway polynomials. They ensure a certain compatibility between the representation of a field and the representations of its subfields.

In the next sections, we will show how the general construction method outlined above works for small finite fields.

Field with four elements
The smallest non-prime field is the field with four elements, which is commonly denoted $GF(4)$ or $$\mathbb{F}_4.$$ It consists of the four elements $0, 1, α, 1 + α$ such that $α^{2} = 1 + α$, $1 ⋅ α = α ⋅ 1 = α$, $x + x = 0$, and $x ⋅ 0 = 0 ⋅ x = 0$, for every $x ∈ GF(4)$, the other operation results being easily deduced from the distributive law. See below for the complete operation tables.

This may be deduced as follows from the results of the preceding section.

Over $GF(2)$, there is only one irreducible polynomial of degree $2$: $$X^2+X+1$$ Therefore, for $GF(4)$ the construction of the preceding section must involve this polynomial, and $$\mathrm{GF}(4) = \mathrm{GF}(2)[X]/(X^2+X+1).$$

Let $α$ denote a root of this polynomial in $GF(4)$. This implies that

and that $α^{2} = 1 + α$ and $α$ are the elements of $1 + α$ that are not in $GF(4)$. The tables of the operations in $GF(2)$ result from this, and are as follows: A table for subtraction is not given, because subtraction is identical to addition, as is the case for every field of characteristic 2. In the third table, for the division of $GF(4)$ by $x + y$, the values of $x⋅y$ must be read in the left column, and the values of $x/y$ in the top row. (Because $x$ for every $z$ in every ring the division by 0 has to remain undefined.) From the tables, it can be seen that the additive structure of $y$ is isomorphic to the Klein four-group, while the non-zero multiplicative structure is isomorphic to the group Z3.

The map $$ \varphi:x \mapsto x^2$$ is the non-trivial field automorphism, called the Frobenius automorphism, which sends $0$ into the second root $1$ of the above mentioned irreducible polynomial $α$.

GF(p2) for an odd prime p
For applying the above general construction of finite fields in the case of $1 + α$, one has to find an irreducible polynomial of degree 2. For $0$, this has been done in the preceding section. If $0$ is an odd prime, there are always irreducible polynomials of the form $1$, with $α$ in $1 + α$.

More precisely, the polynomial $1$ is irreducible over $1$ if and only if $0$ is a quadratic non-residue modulo $1 + α$ (this is almost the definition of a quadratic non-residue). There are $α$ quadratic non-residues modulo $α$. For example, $α$ is a quadratic non-residue for $1 + α$, and $0$ is a quadratic non-residue for $1$. If $1 + α$, that is $1 + α$, one may choose $α$ as a quadratic non-residue, which allows us to have a very simple irreducible polynomial $1$.

Having chosen a quadratic non-residue $0$, let $x$ be a symbolic square root of $y$, that is, a symbol that has the property $0$, in the same way that the complex number $1$ is a symbolic square root of $α$. Then, the elements of $1 + α$ are all the linear expressions $$a+b\alpha,$$ with $0$ and $0$ in $0$. The operations on $0$ are defined as follows (the operations between elements of $0$ represented by Latin letters are the operations in $1$): $$\begin{align} -(a+b\alpha)&=-a+(-b)\alpha\\ (a+b\alpha)+(c+d\alpha)&=(a+c)+(b+d)\alpha\\ (a+b\alpha)(c+d\alpha)&=(ac + rbd)+ (ad+bc)\alpha\\ (a+b\alpha)^{-1}&=a(a^2-rb^2)^{-1}+(-b)(a^2-rb^2)^{-1}\alpha \end{align}$$

GF(8) and GF(27)
The polynomial $$X^3-X-1$$ is irreducible over $0$ and $1$, that is, it is irreducible modulo $α$ and $1 + α$ (to show this, it suffices to show that it has no root in $α$ nor in $0$). It follows that the elements of $α$ and $1 + α$ may be represented by expressions $$a+b\alpha+c\alpha^2,$$ where $1$ are elements of $1 + α$ or $0$ (respectively), and $1 + α$ is a symbol such that $$\alpha^3=\alpha+1.$$

The addition, additive inverse and multiplication on $1$ and $α$ may thus be defined as follows; in following formulas, the operations between elements of $x$ or $y$, represented by Latin letters, are the operations in $1$ or $α$, respectively: $$ \begin{align} -(a+b\alpha+c\alpha^2)&=-a+(-b)\alpha+(-c)\alpha^2 \qquad\text{(for } \mathrm{GF}(8), \text{this operation is the identity)}\\ (a+b\alpha+c\alpha^2)+(d+e\alpha+f\alpha^2)&=(a+d)+(b+e)\alpha+(c+f)\alpha^2\\ (a+b\alpha+c\alpha^2)(d+e\alpha+f\alpha^2)&=(ad + bf+ce)+ (ae+bd+bf+ce+cf)\alpha+(af+be+cd+cf)\alpha^2 \end{align} $$

GF(16)
The polynomial $$X^4+X+1$$ is irreducible over $1 + α$, that is, it is irreducible modulo $0$. It follows that the elements of $0$ may be represented by expressions $$a+b\alpha+c\alpha^2+d\alpha^3,$$ where $0$ are either $0$ or $1$ (elements of $1$), and $1 + α$ is a symbol such that $$\alpha^4=\alpha+1$$ (that is, $α$ is defined as a root of the given irreducible polynomial). As the characteristic of $α$ is $α$, each element is its additive inverse in $1$. The addition and multiplication on $1 + α$ may be defined as follows; in following formulas, the operations between elements of $1 + α$, represented by Latin letters are the operations in $1 + α$. $$ \begin{align} (a+b\alpha+c\alpha^2+d\alpha^3)+(e+f\alpha+g\alpha^2+h\alpha^3)&=(a+e)+(b+f)\alpha+(c+g)\alpha^2+(d+h)\alpha^3\\ (a+b\alpha+c\alpha^2+d\alpha^3)(e+f\alpha+g\alpha^2+h\alpha^3)&=(ae+bh+cg+df) +(af+be+bh+cg+df +ch+dg)\alpha\;+\\ &\quad\;(ag+bf+ce +ch+dg+dh)\alpha^2 +(ah+bg+cf+de +dh)\alpha^3 \end{align} $$

The field $α$ has eight primitive elements (the elements that have all nonzero elements of $1$ as integer powers). These elements are the four roots of $x$ and their multiplicative inverses. In particular, $y$ is a primitive element, and the primitive elements are $x$ with $y$ less than and coprime with $0 ⋅ z = 0$ (that is, 1, 2, 4, 7, 8, 11, 13, 14).

Multiplicative structure
The set of non-zero elements in $GF(4)$ is an abelian group under the multiplication, of order $α$. By Lagrange's theorem, there exists a divisor $1 + α$ of $X^{2} + X + 1$ such that $GF(p^{2})$ for every non-zero $p = 2$ in $p$. As the equation $X^{2} − r$ has at most $r$ solutions in any field, $GF(p)$ is the lowest possible value for $X^{2} − r$. The structure theorem of finite abelian groups implies that this multiplicative group is cyclic, that is, all non-zero elements are powers of a single element. In summary:

Such an element $GF(p)$ is called a primitive element of $r$. Unless $p$, the primitive element is not unique. The number of primitive elements is $p − 1⁄2$ where $p$ is Euler's totient function.

The result above implies that $2$ for every $p = 3, 5, 11, 13, ...$ in $3$. The particular case where $p = 5, 7, 17, ...$ is prime is Fermat's little theorem.

Discrete logarithm
If $p ≡ 3 mod 4$ is a primitive element in $p = 3, 7, 11, 19, ...$, then for any non-zero element $−1 ≡ p − 1$ in $X^{2} + 1$, there is a unique integer $r$ with $α$ such that

This integer $r$ is called the discrete logarithm of $α^{2} = r$ to the base $i$.

While $−1$ can be computed very quickly, for example using exponentiation by squaring, there is no known efficient algorithm for computing the inverse operation, the discrete logarithm. This has been used in various cryptographic protocols, see Discrete logarithm for details.

When the nonzero elements of $GF(p^{2})$ are represented by their discrete logarithms, multiplication and division are easy, as they reduce to addition and subtraction modulo $a$. However, addition amounts to computing the discrete logarithm of $b$. The identity

allows one to solve this problem by constructing the table of the discrete logarithms of $GF(p)$, called Zech's logarithms, for $GF(p^{2})$ (it is convenient to define the discrete logarithm of zero as being $GF(p)$).

Zech's logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table of the same size as the order of the field.

Roots of unity
Every nonzero element of a finite field is a root of unity, as $GF(p)$ for every nonzero element of $GF(2)$.

If $GF(3)$ is a positive integer, an $2$th primitive root of unity is a solution of the equation $3$ that is not a solution of the equation $GF(2)$ for any positive integer $GF(3)$. If $GF(8)$ is a $GF(27)$th primitive root of unity in a field $a, b, c$, then $GF(2)$ contains all the $GF(3)$ roots of unity, which are $α$.

The field $GF(8)$ contains a $GF(27)$th primitive root of unity if and only if $GF(2)$ is a divisor of $GF(3)$; if $GF(2)$ is a divisor of $GF(3)$, then the number of primitive $GF(2)$th roots of unity in $2$ is $GF(16)$ (Euler's totient function). The number of $a, b, c, d$th roots of unity in $0$ is $1$.

In a field of characteristic $GF(2)$, every $α$th root of unity is also a $α$th root of unity. It follows that primitive $GF(2)$th roots of unity never exist in a field of characteristic $2$.

On the other hand, if $GF(16)$ is coprime to $GF(16)$, the roots of the $GF(2)$th cyclotomic polynomial are distinct in every field of characteristic $GF(2)$, as this polynomial is a divisor of $GF(16)$, whose discriminant $GF(16)$ is nonzero modulo $X^{4} + X + 1$. It follows that the $α$th cyclotomic polynomial factors over $α^{m}$ into distinct irreducible polynomials that have all the same degree, say $m$, and that $15$ is the smallest field of characteristic $GF(q)$ that contains the $q – 1$th primitive roots of unity.

Example: GF(64)
The field $k$ has several interesting properties that smaller fields do not share: it has two subfields such that neither is contained in the other; not all generators (elements with minimal polynomial of degree $q – 1$ over $x^{k} = 1$) are primitive elements; and the primitive elements are not all conjugate under the Galois group.

The order of this field being $x$, and the divisors of $GF(q)$ being $x^{k} = 1$, the subfields of $k$ are $q – 1$, $k$, $GF(q)$, and $a$ itself. As $q – 1$ and $GF(q)$ are coprime, the intersection of $a, a^{2}, ..., a^{q−2}, a^{q−1} = 1$ and $a$ in $GF(q)$ is the prime field $q = 2, 3$.

The union of $φ(q − 1)$ and $φ$ has thus $x^{q} = x$ elements. The remaining $x$ elements of $GF(q)$ generate $q$ in the sense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree $a$ over $GF(q)$. This implies that, over $x$, there are exactly $F$ irreducible monic polynomials of degree $n$. This may be verified by factoring $0 ≤ n ≤ q − 2$ over $x = a^{n}$.

The elements of $n$ are primitive $x$th roots of unity for some $a$ dividing $a^{n}$. As the 3rd and the 7th roots of unity belong to $GF(q)$ and $q – 1$, respectively, the $a^{m} + a^{n}$ generators are primitive $a^{m} + a^{n} = a^{n}(a^{m−n} + 1)$th roots of unity for some $a^{n} + 1$ in $n = 0, ..., q − 2$. Euler's totient function shows that there are $−∞$ primitive $x^{q−1} = 1$th roots of unity, $GF(q)$ primitive $n$st roots of unity, and $n$ primitive $x^{n} = 1$rd roots of unity. Summing these numbers, one finds again $x^{m} = 1$ elements.

By factoring the cyclotomic polynomials over $m < n$, one finds that:
 * The six primitive $a$th roots of unity are roots of $$X^6+X^3+1,$$ and are all conjugate under the action of the Galois group.
 * The twelve primitive $n$st roots of unity are roots of $$(X^6+X^4+X^2+X+1)(X^6+X^5+X^4+X^2+1).$$ They form two orbits under the action of the Galois group. As the two factors are reciprocal to each other, a root and its (multiplicative) inverse do not belong to the same orbit.
 * The $F$ primitive elements of $F$ are the roots of $$(X^6+X^4+X^3+X+1)(X^6+X+1)(X^6+X^5+1)(X^6+X^5+X^3+X^2+1)(X^6+X^5+X^2+X+1)(X^6+X^5+X^4+X+1).$$ They split into six orbits of six elements each under the action of the Galois group.

This shows that the best choice to construct $n$ is to define it as $1, a, a^{2}, ..., a^{n−1}$. In fact, this generator is a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division.

Frobenius automorphism and Galois theory
In this section, $GF(q)$ is a prime number, and $n$ is a power of $n$.

In $q − 1$, the identity $n$ implies that the map $$ \varphi:x \mapsto x^p$$ is a $q − 1$-linear endomorphism and a field automorphism of $n$, which fixes every element of the subfield $GF(q)$. It is called the Frobenius automorphism, after Ferdinand Georg Frobenius.

Denoting by $φ(n)$ the composition of $n$ with itself $GF(q)$ times, we have $$ \varphi^k:x \mapsto x^{p^k}.$$ It has been shown in the preceding section that $gcd(n, q − 1)$ is the identity. For $p$, the automorphism $(np)$ is not the identity, as, otherwise, the polynomial $$X^{p^k}-X$$ would have more than $n$ roots.

There are no other $(np)$-automorphisms of $p$. In other words, $n$ has exactly $p$ $n$-automorphisms, which are $$\mathrm{Id}=\varphi^0, \varphi, \varphi^2, \ldots, \varphi^{n-1}.$$

In terms of Galois theory, this means that $p$ is a Galois extension of $X^{n} − 1$, which has a cyclic Galois group.

The fact that the Frobenius map is surjective implies that every finite field is perfect.

Polynomial factorization
If $n^{n}$ is a finite field, a non-constant monic polynomial with coefficients in $p$ is irreducible over $n$, if it is not the product of two non-constant monic polynomials, with coefficients in $GF(p)$.

As every polynomial ring over a field is a unique factorization domain, every monic polynomial over a finite field may be factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials.

There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite field. They are a key step for factoring polynomials over the integers or the rational numbers. At least for this reason, every computer algebra system has functions for factoring polynomials over finite fields, or, at least, over finite prime fields.

Irreducible polynomials of a given degree
The polynomial $$X^q-X$$ factors into linear factors over a field of order $d$. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order $GF(p^{d})$.

This implies that, if $p$ then $n$ is the product of all monic irreducible polynomials over $GF(64)$, whose degree divides $6$. In fact, if $GF(2)$ is an irreducible factor over $2^{6}$ of $6$, its degree divides $1, 2, 3, 6$, as its splitting field is contained in $GF(64)$. Conversely, if $GF(2)$ is an irreducible monic polynomial over $GF(2^{2}) = GF(4)$ of degree $GF(2^{3}) = GF(8)$ dividing $GF(64)$, it defines a field extension of degree $2$, which is contained in $3$, and all roots of $GF(4)$ belong to $GF(8)$, and are roots of $GF(64)$; thus $GF(2)$ divides $GF(4)$. As $GF(8)$ does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it.

This property is used to compute the product of the irreducible factors of each degree of polynomials over $10$; see Distinct degree factorization.

Number of monic irreducible polynomials of a given degree over a finite field
The number $54$ of monic irreducible polynomials of degree $GF(64)$ over $GF(64)$ is given by $$N(q,n)=\frac{1}{n}\sum_{d\mid n} \mu(d)q^{n/d},$$ where $6$ is the Möbius function. This formula is an immediate consequence of the property of $GF(2)$ above and the Möbius inversion formula.

By the above formula, the number of irreducible (not necessarily monic) polynomials of degree $GF(2)$ over $9 = 54⁄6$ is $6$.

The exact formula implies the inequality $$N(q,n)\geq\frac{1}{n} \left(q^n-\sum_{\ell\mid n, \ \ell \text{ prime}} q^{n/\ell}\right);$$ this is sharp if and only if $X^{64} − X$ is a power of some prime. For every $GF(2)$ and every $GF(64)$, the right hand side is positive, so there is at least one irreducible polynomial of degree $n$ over $n$.

Applications
In cryptography, the difficulty of the discrete logarithm problem in finite fields or in elliptic curves is the basis of several widely used protocols, such as the Diffie–Hellman protocol. For example, in 2014, a secure internet connection to Wikipedia involved the elliptic curve Diffie–Hellman protocol (ECDHE) over a large finite field. In coding theory, many codes are constructed as subspaces of vector spaces over finite fields.

Finite fields are used by many error correction codes, such as Reed–Solomon error correction code or BCH code. The finite field almost always has characteristic of $63$, since computer data is stored in binary. For example, a byte of data can be interpreted as an element of $GF(4)$. One exception is PDF417 bar code, which is $GF(8)$. Some CPUs have special instructions that can be useful for finite fields of characteristic $54$, generally variations of carry-less product.

Finite fields are widely used in number theory, as many problems over the integers may be solved by reducing them modulo one or several prime numbers. For example, the fastest known algorithms for polynomial factorization and linear algebra over the field of rational numbers proceed by reduction modulo one or several primes, and then reconstruction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.

Similarly many theoretical problems in number theory can be solved by considering their reductions modulo some or all prime numbers. See, for example, Hasse principle. Many recent developments of algebraic geometry were motivated by the need to enlarge the power of these modular methods. Wiles' proof of Fermat's Last Theorem is an example of a deep result involving many mathematical tools, including finite fields.

The Weil conjectures concern the number of points on algebraic varieties over finite fields and the theory has many applications including exponential and character sum estimates.

Finite fields have widespread application in combinatorics, two well known examples being the definition of Paley Graphs and the related construction for Hadamard Matrices. In arithmetic combinatorics finite fields and finite field models are used extensively, such as in Szemerédi's theorem on arithmetic progressions.

Wedderburn's little theorem
A division ring is a generalization of field. Division rings are not assumed to be commutative. There are no non-commutative finite division rings: Wedderburn's little theorem states that all finite division rings are commutative, and hence are finite fields. This result holds even if we relax the associativity axiom to alternativity, that is, all finite alternative division rings are finite fields, by the Artin–Zorn theorem.

Algebraic closure
A finite field $n$ is not algebraically closed: the polynomial $$f(T) = 1+\prod_{\alpha \in F} (T-\alpha),$$ has no roots in $n$, since ${9, 21, 63}$ for all $6$ in $9$.

Given a prime number $p$, let $$\overline{\mathbb{F}}_p$$ be an algebraic closure of $$\mathbb{F}_p.$$ It is not only unique up to an isomorphism, as do all algebraic closures, but contrarily to the general case, all its subfield are fixed by all its automorphisms, and it is also the algebraic closure of all finite fields of the same characteristic $p$.

This property results mainly from the fact that the elements of $$\mathbb F_{p^n}$$ are exactly the roots of $$x^{p^n}-x,$$ and this defines an inclusion $$\mathbb \mathbb F_{p^n}\subset \mathbb F_{p^{nm}} $$ for $$m>1.$$ These inclusions allow writing informally $$\overline{\mathbb{F}}_p = \bigcup_{n \ge 1} \mathbb{F}_{p^n}.$$ The formal validation of this notation results from the fact that the above field inclusions form a directed set of fields; Its direct limit is $$\overline{\mathbb{F}}_p,$$ which may thus be considered as "directed union".

Primitive elements in the algebraic closure
Given a primitive element $$g_{mn}$$ of $$\mathbb{F}_{q^{mn}},$$ then $$g_{mn}^m$$ is a primitive element of $$\mathbb{F}_{q^{n}}.$$

For explicit computations, it may be useful to have a coherent choice of the primitive elements for all finite fields; that is, to choose the primitive element $$g_{n}$$ of $$\mathbb{F}_{q^{n}}$$ in order that, whenever $$n=mh,$$ one has $$g_{m}=g_n^h,$$ where $$g_{m}$$ is the primitive element already chosen for $$\mathbb{F}_{q^{m}}.$$

Such a construction may be obtained by Conway polynomials.

Quasi-algebraic closure
Although finite fields are not algebraically closed, they are quasi-algebraically closed, which means that every homogeneous polynomial over a finite field has a non-trivial zero whose components are in the field if the number of its variables is more than its degree. This was a conjecture of Artin and Dickson proved by Chevalley (see Chevalley–Warning theorem).

title: Finite field


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Franchising

Franchising is based on a marketing concept which can be adopted by an organization as a strategy for business expansion. Where implemented, a franchisor licenses some or all of its know-how, procedures, intellectual property, use of its business model, brand, and rights to sell its branded products and services to a franchisee. In return, the franchisee pays certain fees and agrees to comply with certain obligations, typically set out in a franchise agreement.

The word franchise is of Anglo-French derivation—from franc, meaning 'free'—and is used both as a noun and as a (transitive) verb. For the franchisor, use of a franchise system is an alternative business growth strategy, compared to expansion through corporate owned outlets or "chain stores". Adopting a franchise system business growth strategy for the sale and distribution of goods and services minimizes the franchisor's capital investment and liability risk.

Franchising is rarely an equal partnership, especially in the typical arrangement where the franchisee is an individual, unincorporated partnership or small privately held corporation, as this will ensure the franchisor has substantial legal and/or economic advantages over the franchisee. The usual exception to this rule is when the prospective franchisee is also a powerful corporate entity controlling a highly lucrative location and/or captive market (for example, a large sports stadium) in which prospective franchisors must then compete to exclude one another from. However, under specific circumstances like transparency, favourable legal conditions, financial means and proper market research, franchising can be a vehicle of success for both a large franchisor and a small franchisee.

Thirty-six countries have laws that explicitly regulate franchising, with the majority of all other countries having laws which have a direct or indirect effect on franchising. Franchising is also used as a foreign market entry mode.

History
The boom in franchising did not take place until after World War II. Nevertheless, the rudiments of modern franchising date back to the Middle Ages when landowners made franchise-like agreements with tax collectors, who retained a percentage of the money they collected and turned the rest over. The practice ended around 1562 but spread to other endeavors. For example, in 17th-century England franchisees were granted the right to sponsor markets and fairs or operate ferries. There was little growth in franchising, though, until the mid-19th century, when it appeared in the United States for the first time.

One of the first successful American franchising operations was started by an enterprising druggist named John S. Pemberton. In 1886, he concocted a beverage comprising sugar, molasses, spices, and cocaine. Pemberton licensed selected people to bottle and sell the drink, which was an early version of what is now known as Coca-Cola. His was one of the earliest—and most successful—franchising operations in the United States.

The Singer Company implemented a franchising plan in the 1850s to distribute its sewing machines. The operation failed, though, because the company did not earn much money even though the machines sold well. The dealers, who had exclusive rights to their territories, absorbed most of the profits because of deep discounts. Some failed to push Singer products, so competitors were able to outsell the company. Under the existing contract, Singer could neither withdraw rights granted to franchisees nor send in its own salaried representatives. So, the company started repurchasing the rights it had sold. The experiment proved to be a failure. That may have been one of the first times a franchisor failed, but it was by no means the last. (Even Colonel Sanders did not initially succeed in his KFC franchising efforts.) Still, the Singer venture did not put an end to franchising.

Other companies tried franchising in one form or another after the Singer experience. For example, several decades later, General Motors established a somewhat successful franchising operation in order to raise capital. Perhaps the father of modern franchising, though, is Louis K. Liggett. In 1902, Liggett invited a group of druggists to join a "drug cooperative". As he explained to them, they could increase profits by paying less for their purchases, especially if they set up their own manufacturing company. His idea was to market private label products. About 40 druggists pooled $4,000 of their own money and adopted the name Rexall. Sales soared, and Rexall became a franchisor. The chain's success set a pattern for other franchisors to follow.

Although many business owners did affiliate with cooperative ventures of one type or another, there was little growth in franchising until the early 20th century, and in whatever form franchising existed, it looked nothing like what it is today. As the United States shifted from an agricultural to an industrial economy, manufacturers licensed individuals to sell automobiles, trucks, gasoline, beverages, and a variety of other products. The franchisees did little more than selling the products, though. The sharing of responsibility associated with contemporary franchising arrangement did not exist to a great extent. Consequently, franchising was not a growth industry in the United States.

It was not until the 1960s and 1970s that people began to take a close look at the attractiveness of franchising. The concept intrigued people with entrepreneurial spirit. However, there were serious pitfalls for investors, which almost ended the practice before it became truly popular.

The United States is a leader in franchising, a position it has held since the 1930s when it used the approach for fast-food restaurants, food inns and, slightly later, motels at the time of the Great Depression. As of 2005, there were 909,253 established franchised businesses, generating $880.9 billion of output and accounting for 8.1 percent of all private, non-farm jobs. This amounts to 11 million jobs, and 4.4 percent of all private sector output.

Mid-sized franchises like restaurants, gasoline stations and trucking stations involve substantial investment and require all the attention of a businessperson.

There are also large franchises like hotels, spas and hospitals, which are discussed further under technological alliances.

"No poaching" agreements are prevalent within franchises, thus limiting the ability of employers at one franchise establishment to hire employees at an affiliated franchise. Economists have characterized these agreements as a contributor to oligopsony.

Fees and contract arrangement
Three important payments are made to a franchisor: (a) a royalty for the trademark, (b) reimbursement for the training and advisory services given to the franchisee, and (c) a percentage of the individual business unit's sales. These three fees may be combined in a single 'management' fee. A fee for "disclosure" is separate and is always a "front-end fee".

A franchise usually lasts for a fixed time period (broken down into shorter periods, which each require renewal), and serves a specific territory or geographical area surrounding its location. One franchisee may manage several such locations. Agreements typically last from five to thirty years, with premature cancellations or terminations of most contracts bearing serious consequences for franchisees. A franchise is merely a temporary business investment involving renting or leasing an opportunity, not the purchase of a business for the purpose of ownership. It is classified as a wasting asset due to the finite term of the license.

Franchise fees are on average 6.7% with an additional average marketing fee of 2%. However, not all franchise opportunities are the same and many franchise organizations are pioneering new models that challenge antiquated structures and redefine success for the organization as well as the franchisee.

A franchise can be exclusive, non-exclusive or "sole and exclusive".

Although franchisor revenues and profit may be listed in a franchise disclosure document (FDD), no laws require an estimate of franchisee profitability, which depends on how intensively the franchisee "works" the franchise. Therefore, franchisor fees are typically based on "gross revenue from sales" and not on profits realized. See remuneration.

Various tangibles and intangibles such as national or international advertising, training and other support services are commonly made available by the franchisor.

Franchise brokers help franchisors find appropriate franchisees. There are also main 'master franchisors' who obtain the rights to sub-franchise in a territory.

According to the International Franchise Association approximately 44% of all businesses in the United States are franchisee-worked.

Rationale and risk shift
Franchising is one of the few means available to access venture capital without the need to give up control of the operation of the chain and build a distribution system for servicing it. After the brand and formula are carefully designed and properly executed, franchisors are able to sell franchises and expand rapidly across countries and continents using the capital and resources of their franchisees while reducing their own risk.

There is also risk for the people buying the franchises. However, failure rates are much lower for franchise businesses than independent business startups.

Franchisor rules imposed by the franchising authority are becoming increasingly strict. Some franchisors are using minor rule violations to terminate contracts and seize the franchise without any reimbursement.

Advantages and disadvantages of franchising as an entry mode
Franchising brings with it several advantages and disadvantages for firms looking to expand into new areas and foreign markets. The primary advantage is that the firm does not have to bear the development cost and risks of opening a foreign market on its own, as the franchisee is typically responsible for those costs and risks, putting the onus on them to build a profitable operation as quickly as possible. Through franchising, a firm has the potential of building a global presence quickly and also at a low cost and risk.

For the franchisee, the primary advantages are access to a well-known brand, support in setting up the business using operating manuals, and ongoing operational support including access to suppliers and employee training.

A primary disadvantage to franchising is quality control, as the franchisor wants the firm's brand name to convey a message to consumers about the quality and consistency of the firm's product. They want the consumer to experience the same quality regardless of location or franchise status. This can prove to be an issue with franchising, as a customer who had a bad experience at one franchise may assume that they will have the same experience at other locations with other services. Distance can make it difficult for firms to detect whether or not the franchises are of poor quality. One way around this disadvantage is to set up extra subsidiaries in each country or state in which the firm expands. This creates a smaller number of franchisees to oversee, which will reduce the quality control challenges.

Obligations of the parties
Each party to a franchise has several interests to protect. The franchisor is involved in securing protection for the trademark, controlling the business concept and securing know-how. The franchisee is obligated to carry out the services for which the trademark has been made prominent or famous. There is a great deal of standardization required. The place of service has to bear the franchisor's signs, logos and trademark in a prominent place. The uniforms worn by the staff of the franchisee have to be of a particular design and color. The service has to be in accordance with the pattern followed by the franchisor in the successful franchise operations. Thus, franchisees are not in full control of the business, as they would be in retailing.

A service can be successful if equipment and supplies are purchased at a fair price from the franchisor or sources recommended by the franchisor. A coffee brew, for example, can be readily identified by the trademark if its raw materials come from a particular supplier. If the franchisor requires purchase from her stores, it may come under anti-trust legislation or equivalent laws of other countries. So, too, with purchases such as the uniforms of personnel and signs, as well as the franchise sites, if they are owned or controlled by the franchisor.

The franchisee must carefully negotiate the license and must develop a marketing or business plan with the franchisor. The fees must be fully disclosed and there should not be any hidden fees. The start-up costs and working capital must be known before the license is granted. There must be assurance that additional licensees will not crowd the "territory" if the franchise is worked according to plan. The franchisee must be seen as an independent merchant. It must be protected by the franchisor from any trademark infringement by third parties. A franchise attorney is required to assist the franchisee during negotiations.

Often the training period – the costs of which are in great part covered by the initial fee – is too short in cases where it is necessary to operate complicated equipment, and the franchisee has to learn on their own from instruction manuals. The training period must be adequate, but in low-cost franchises it may be considered expensive. Many franchisors have set up corporate universities to train staff online. This is in addition to providing literature, sales documents and email access.

Also, franchise agreements carry no guarantees or warranties and the franchisee has little or no recourse to legal intervention in the event of a dispute. Franchise contracts tend to be unilateral and favor of the franchisor, who is generally protected from lawsuits from their franchisees because of the non-negotiable contracts that franchisees are required to acknowledge, in effect, that they are buying the franchise knowing that there is risk, and that they have not been promised success or profits by the franchisor. Contracts are renewable at the sole option of the franchisor. Most franchisors require franchisees to sign agreements that mandate where and under what law any dispute would be litigated.

Australia
In 2016 there were an estimated 1,120 franchise brands operating in Australia and an estimated 79,000 units operating in business format franchises, with a total brand turnover of approximately $146 billion and a sales revenue of approximately $66.5 billion. In 2016 the majority of franchise brands were retailers with the largest segment being non-food retailing, accounting for 26 percent of brands, a further 19 percent of brands were involved in food retailing, 15 percent of franchisors operated in administration and support services, 10 percent in other services, 7 percent in education and training and 7 percent in rental, hire and real estate services.

Franchising in Australia commenced in a significant way in the early 1970s under the influence of the franchised US fast food systems such as KFC, Pizza Hut, and McDonald's. It was however underway prior to this and a decade earlier in 1960 Leslie Joseph Hooker, considered a pioneer of franchising, created Australia's first national real estate agency network of Hooker real estate agencies.

In Australia, franchising is regulated by the Franchising Code of Conduct, a mandatory code of conduct concluded under the Trade Practices Act 1974. The ACCC regulates the Franchising Code of Conduct, which is a mandatory industry code that applies to the parties to a franchise agreement. This code requires franchisors to produce a disclosure document which must be given to a prospective franchisee at least 14 days before the franchise agreement is entered into.

The code also regulates the content of franchise agreements, for example in relation to marketing funds, a cooling-off period, termination, and the resolution of disputes by mediation.

Franchising code of conduct
On 1 January 2015, the old Franchising Code was repealed and replaced with a new Franchising Code of Conduct. The new Code applies to conduct on or after 1 January 2015.

The new Code:
 * introduces an obligation under the Code for parties to act in good faith in their dealings with one another
 * introduces financial penalties and infringement notices for serious breaches of the Code
 * requires franchisors to provide prospective franchisees with a short information sheet outlining the risks and rewards of franchising
 * requires franchisors to provide greater transparency in the use of and accounting for money used for marketing and advertising and to set up a separate marketing fund for marketing and advertising fees
 * requires additional disclosure about the ability of the franchisor and a franchisee to sell online
 * prohibits franchisors from imposing significant capital expenditure except in limited circumstances.

These are significant changes and it is important that franchisors, franchisees and potential franchises understand their rights and responsibilities under the Code.

For further information about the changes to the Code, please see the updated Franchisor Compliance Manual and the Franchisee Manual.

The Code explanatory materials are available from the ComLaw website (link is external).

New Zealand
New Zealand is served by around 423 franchise systems operating 450 brands, giving it the highest proportion of franchises per capita in the world. Despite (or because of) the 2008–2009 recession, the total number of franchised units increased by 5.3% from 2009 to 2010. There is no separate law covering franchises, so they are covered by normal commercial law. This functions very well in New Zealand and includes law as it applies to contracts, restrictive trade practices, intellectual property, and the law of misleading or deceptive conduct.

The Franchise Association of New Zealand introduced a self-regulatory code of practice for its members in 1996. This contains many provisions similar to those of the Australian Franchising Code of Practice legislation, although only around a third of all franchises are members of the association and therefore bound by the code.

A case of fraud in 2007 perpetrated by a former master franchisee of the country's largest franchise system led to a review of the need for franchise law by the Ministry of Economic Development. The New Zealand Government decided there was no case for franchise-specific legislation at that time. This decision was criticised by the opposition, which had initiated the review when in power, and the review process was questioned by a leading academic. The Franchise Association originally supported the positive regulation of the franchise sector but its eventual submission to the review was in favour of the status quo of self-regulation.

Brazil
By the end of 2012, about 2,031 franchise brands were operating in Brazil, with approximately 93,000 locations, making it one of the largest countries in the world in terms of number of units. Around 11 percent of this total were foreign-based franchisors.

The Brazilian Franchise Law (Law No. 8955 of December 15, 1994) defines the franchise as a system in which the franchisor licenses the franchisee, for a payment, the right to use a trademark or patent along with the right to distribute products or services on an exclusive or semi-exclusive basis. The provision of a "Franchise Offer Circular", or disclosure document, is mandatory before execution of agreement and is valid for all of the Brazilian territory. Failure to disclose voids the agreement, which leads to refunds and serious payments for damages. The Franchise Law does not distinguish between Brazilian and foreign franchisors. The National Institute of Industrial Property (INPI) is the registering authority. Indispensable documents are a Statement of Delivery (of disclosure documentation) and a Certification of Recording (INPI). The latter is necessary for payments. All sums may not be convertible into foreign currency. Certification may also mean compliance with Brazil's antitrust legislation.

Parties to international franchising may decide to adopt the English language for the document, as long as the Brazilian party knows English fluently and expressly acknowledges that fact, to avoid translation. The registration accomplishes three things:

 * * It make the agreement effective against third parties
 * * It permits the remittance of payments
 * * It qualifies the franchisee for tax deductions.

Canada
In Canada, recent legislation has mandated better disclosure and fair treatment of franchisees. The regulations also ensure their right to form associations and launch collective action, even if they signed contracts prohibiting such moves. Franchising in Canada involves 1,300 brands, 80,000 franchise units accounting for about 20% of all consumer spending.

China
China has the most franchises in the world but the scale of their operations is relatively small. The average franchise system in China has about 45 outlets, compared to more than 540 in the United States. Together, there are 2600 brands in some 200,000 retail markets. KFC was the most significant foreign entry in 1987 and is widespread. Many franchises are in fact joint-ventures, as at their forming the franchise law was not explicit. For example, McDonald's is a joint venture. Pizza Hut, TGIF, Wal-mart, Starbucks followed not long thereafter. But total franchising is only 3% of retail trade, which seeks foreign franchise growth.

The year 2005 saw the birth of an updated franchise law, "Measures for the Administration of Commercial Franchise". Previous legislation (1997) made no specific inclusion of foreign investors. Further updates were made in 2007, with the objective of increased clarity of the law.

The laws are applicable if there are transactions involving a trademark combined with payments with many obligations on the franchisor. The law comprises 42 articles and eight chapters.

Among the franchisor obligations are:
 * The FIE (foreign-invested enterprise) franchisor must be registered by the regulator
 * The franchisor (or its subsidiary) must have operated at least two company-owned franchises in China (revised to "anywhere") for more than 12 months ("the two-shop, one-year" rule)
 * The franchisor must disclose any information requested by the franchisee
 * Cross-border franchising, with some caveats, is possible (2007 law).

The franchisor must meet a list of requirements for registration, among which are:
 * The standard franchise agreement, working manual and working capital requirements,
 * A track-record of operations, and ample ability to supply materials,
 * The ability to train the Chinese personnel and provide long-term operational guidance,
 * The franchise agreement must have a minimum three-year term.

Among other provisions:
 * The franchisor is liable for certain actions of its suppliers
 * Monetary and other penalties apply for infractions of the regulations.

The disclosure must take place 20 days in advance. It has to contain:
 * Details of the franchisor's experience in the franchised business with scope of business
 * Identification of the franchisor's principal officers
 * Litigation of the franchisor during the past five years
 * Full details about all franchise fees
 * The amount of a franchisee's initial investment
 * A list of the goods or services the franchisor can supply, and the terms of supply
 * The training franchisees will receive
 * Information about the trademarks, including registration, usage and litigation
 * Demonstration of the franchisor's capabilities to provide training and guidance
 * Statistics about existing units, including number, locations and operational results, and the percentage of franchises that have been terminated, and
 * An audited financial report and tax information (for an unspecified period of time).

Other elements of this legislation are:
 * The franchisee's confidentiality obligations continue indefinitely after termination or expiration of the franchise agreement
 * If the franchisee has paid a deposit to the franchisor, it must be refunded on termination of the franchise agreement; upon termination, the franchisee is prohibited from continuing to use the franchisor's marks.

India
The franchising of foreign goods and services to India is in its infancy. The first International Exhibition was only held in 2009. India is, however, one of the biggest franchising markets because of its large middle-class of 300 million who are not reticent about spending and because the population is entrepreneurial in character. In a highly diversified society, (see Demographics of India) McDonald's is a success story despite its menu differing from that of the rest of the world.

So far, franchise agreements are covered under two standard commercial laws: the Contract Act 1872 and the Specific Relief Act 1963, which provide for both specific enforcement of covenants in a contract and remedies in the form of damages for breach of contract.

Kazakhstan
In Kazakhstan franchise turnover for 2013 is US$2.5 billion per year. Kazakhstan is the leader in Central Asia in the franchising market. A special law on franchising came into effect in 2002. There are more than 300 franchise systems and the number of franchised outlets approaches 2000. Kazakhstan franchising began with the emergence of a Coca-Cola factory, opened to sublicense a Turkish licensor of the same brand. The plant was built in 1994. Other brands that are also present in Kazakhstan through the franchise system include Pepsi, Hilton, Marriott, Intercontinental, and Pizza Hut.

Europe
Franchising has grown rapidly in Europe in recent years, but the industry is largely unregulated. The European Union has not adopted a uniform franchise law. Only six of the 27 member states have a pre-contract disclosure law. They are France (1989), Spain (1996), Romania (1997), Italy (2004), Sweden (2004) and Belgium (2005). Estonia and Lithuania have franchise laws that impose mandatory terms on franchise agreements. In Spain there is also mandatory registration on a public registry. Although they have no franchise specific laws, Germany and those countries with a legal system based upon that of Germany, such as Austria, Greece and Portugal, probably impose the greatest regulatory burden on franchisors due to their tendency to treat franchisees as quasi consumers in certain circumstances and the willingness of the judiciary to use the concept of good faith to make pro-franchisee decisions. In the UK, the recent Papa John case shows that there is also a need for pre-contractual disclosure and the Yam Seng case shows that there is a duty of good faith in franchise relationships.

The European Franchising Federation's Code of Ethics has been adopted by seventeen national franchise associations. However this has no legal force and enforcement by the national associations is neither uniform of rigorous. Commentators like Dr Mark Abell, in his book The Law and Regulation of Franchising in the EU (published in 2013 by Edward Elgar, ISBN 978 1 78195 2207) consider this lack of uniformity to be one of the greatest barriers to franchising realising its potential in the EU.

When adopting a European strategy, it is important that a franchisor takes expert legal advice. Most often one of the principal tasks in Europe is to find retail space, which is not so significant a factor in the US. This is where the franchise broker, or the master franchisor, plays an important role. Cultural factors are also relevant, as local populations tend to be heterogeneous.

France
France is one of Europe's largest markets. Similar to the United States, it has a long history of franchising, dating back to the 1930s. Growth came in the 1970s. The market is considered difficult for outside franchisors because of cultural characteristics, yet McDonald's and Century 21 are found everywhere. There are some 30 U.S. firms involved in franchising in France.

There are no government agencies regulating franchises. The Loi Doubin Law of 1989 was the first European franchise disclosure law. Combined with Decree No. 91-337, it regulates disclosure, although the decree also applies to any person who provides to another person a corporate name, trademark or trade name or other business arrangements. The law applies to "exclusive or quasi-exclusive territory". The disclosure document must be delivered at least 20 days before the execution of the agreement or any payments are made.

The specific and important disclosures to be made are:

 * 1)  The date of the founding of the franchisor's enterprise and a summary of its business history and all information necessary to assess the business experience of the franchisor, including bankers,
 * 2)  A description of the local market for the goods or services,
 * 3)  The franchisor's financial statements for the previous two years,
 * 4)  A list of all other franchisees currently in the network,
 * 5)  All franchisees who have left the network during the preceding year, whether by termination or non-renewal, and
 * 6)  The conditions for renewal, assignment, termination and the scope of exclusivity.

Initially, there was some uncertainty whether any breach of the provisions of the Doubin Law would enable the franchisee to walk away from the contract. However, the French supreme court (Cour de cassation) eventually ruled that agreements should only be annulled where missing or incorrect information affected the decision of the franchisee to enter into the agreement. The burden of proof is on the franchisee.

Dispute settlement features are only incorporated in some European countries. By not being rigorous, franchising is encouraged.

Italy
Under Italian law franchise is defined as an arrangement between two financially independent parties where a franchisee is granted, in exchange for a consideration, the right to market goods and services under particular trademarks. In addition, articles dictate the form and content of the franchise agreement and define the documents that must be made available 30 days prior to execution. The franchisor must disclose:

 * a) A summary of the franchise activities and operations,
 * b) A list of franchisees currently operating in the franchise system in Italy,
 * c) Year-by-year details of the changes in the number of franchisees for the previous three years in Italy,
 * d) A summary of any court or arbitral proceedings in Italy related to the franchise system, and
 * e) If requested by the franchisee, copies of franchisor's balance sheets for the previous three years, or since start-up if that period is shorter.

Norway
There are no specific laws regulating franchising in Norway. However, the Norwegian Competition Act section 10 prohibits cooperation which may prevent, limit or diminish the competition. This may also apply to vertical cooperation such as franchising.

Russia
In Russia, under chapter 54 of the Civil Code (passed 1996), franchise agreements are invalid unless written and registered, and franchisors cannot set standards or limits on the prices of the franchisee's goods. Enforcement of laws and resolution of contractual disputes is a problem: Dunkin' Donuts chose to terminate its contract with Russian franchisees who were selling vodka and meat patties contrary to their contracts, rather than pursue legal remedies.

Spain
The legal definition of franchising in Spain is an activity in which an undertaking, the franchisor, grants to another party, the franchisee, for a specific market and in exchange for financial compensation (either direct, indirect or both), the right to exploit an owned system to commercialize products or services already exploited by the franchisor with enough success and experience.

The Spanish Retail Trading Act regulates franchising. The contents of the franchise must include, at least:
 * The use of a common name or brand or any other intellectual property right and a uniform presentation of the premises or the transport means included in the agreement.
 * The communication by the franchisor to the franchise of certain technical knowledge or substantial and singular know-how that has to be owned by the franchisor, and
 * Technical or commercial assistance or both, provided by the franchisor to the franchisee during the agreement, without prejudice to any supervision faculty to which the parties could freely agree in the contract.

In Spain, the franchisor submits the disclosure information 20 days prior to signing the agreement or prior to any payment made by the franchisee to the franchisor. Franchisors are to disclose to the potential franchisee specific information in writing. This information has to be true and not misleading and include:
 * Identification of the franchisor;
 * Justification of ownership or license for use of any trademark or similar sign and judicial claims affecting them as well as the duration of the license;
 * General description of the sector in which the franchise operates;
 * Experience of the franchisor;
 * Contents and characteristics of the franchise and its exploitation;
 * Structure and extension of the network in Spain;
 * Essential elements of the franchise agreement.

Franchisors (with some exceptions) should be registered in the Franchisors' Register and provide the requested information. According to the regulation in force in 2010 this obligation has to be met within three months after the start of its activities in Spain.

Turkey
Franchising is a sui generis contract which bears the characteristics of several explicitly regulated contracts such as; agency, sales contract and so forth. The regulations concerning these kinds of contracts in Turkish Commercial Code and in Turkish Code of Obligations are applied to franchising. Franchising is described in doctrine and has several essential components such as; the independence of the franchisee from the franchisor, the use of know-how and the uniformity of product and services, standard use of the brand and logo, payment of a royalty fee, increasement of sales by the franchisee and continuity. Franchising may be for a determined or undetermined period of time. The undetermined one can only be annulled either by a notice before a reasonable amount of time or by a just cause. The franchising agreement with a determined time period ends within the end of the time period if not specified otherwise in the agreement. However, termination based on just cause is also foreseen for franchising agreement with a determined time period.

United Kingdom
In the United Kingdom there are no franchise-specific laws, and franchises are subject to the same laws that govern other businesses. Even without direct legislation, judicial decisions indicate that a franchisor is expected to provide a clear disclosure of relevant facts before the franchisee enters into a franchise, and that franchisors have a duty of good faith. The Trading Schemes Act, which governs arrangements in which participants may receive a benefit or reward for introducing other participants to a scheme or sell goods or services provided by the person who is promoting the scheme, may apply to multi-tiered franchises. The industry engages in some self-regulation through the British Franchise Association (BFA) and the Quality Franchise Association (QFA).

There are a number of franchise businesses which are not members of the BFA and many which do not meet the BFA membership criteria. Part of the BFA's role in self-regulation is to work with franchisors through the application process and recommend changes which will lead to the franchise business meeting BFA standards. A number of businesses that refer to themselves as franchises do not conform to the BFA Code of Ethics are therefore excluded from membership.

On 22 May 2007, hearings were held in the UK Parliament concerning citizen-initiated petitions for special regulation of franchising by the government of the UK due to losses incurred by citizens who had invested in franchises. The Minister for Industry and the Regions, Margaret Hodge, conducted hearings but saw no need for any government regulation of franchising with the advice that government regulation of franchising might lull the public into a false sense of security. Mr Mark Prisk MP suggested that the costs of such regulation to the franchisee and franchisor could be prohibitive and would in any case provide a system which mirrored the work already being completed by the BFA. The Minister for Industry and the Regions indicated that if due diligence were performed by the investors and the banks, the current laws governing business contracts in the UK offered sufficient protection for the public and the banks. The debate also made reference to the self-regulatory function performed by the BFA recognizing that the association "punched above its weight".

In the 2010 case of MGB Printing v Kall Kwik UK Ltd., the High Court established that a franchisor may assume a duty of care to a franchisee in certain circumstances. Kall Kwik, a design and print franchisor, had incorrectly advised MGB, who was purchasing a franchise, of the costs of undertaking refit work needed to meet Kall Kwik's franchising requirements. In this particular case, Kall Kwik had stated that they would provide professional advice to potential franchisees, and because they had not provided details of the fitting standards which must be met, they had encouraged MGB to rely on the advice offered by themselves.

On 3 June 2021, it was announced that the Approved Franchise Association (AFA) would merge with the British Franchise Association (BFA) and that both franchise associations would operate under the BFA umbrella.

United States
Isaac Singer, who made improvements to an existing model of a sewing machine in the 1850s, began one of the first franchising efforts in the United States, followed later by Coca-Cola, Western Union, and by agreements between automobile manufacturers and dealers.

Modern franchising came to prominence with the rise of franchise-based food service establishments. In 1932, Howard Deering Johnson established the first modern restaurant franchise based on his successful Quincy, Massachusetts Howard Johnson's restaurant founded in the late 1920s. The idea was to let independent operators use the same name, food, supplies, logo and even building design in exchange for a fee. The growth in franchising accelerated in the 1930s when such chains as Howard Johnson's started to franchise motels. The 1950s saw a boom in franchise chains in conjunction with the development of the U.S. Interstate Highway System and the growing popularity of fast food.

The Federal Trade Commission has oversight of franchising via the FTC Franchise Rule.

The FTC requires that the franchisee be furnished with a Franchise Disclosure Document (FDD) by the franchisor at least fourteen days before money changes hands or a franchise agreement is signed. Whereas elements of the disclosure may be available from third parties, only that provided by the franchisor can be depended upon. The U.S. Franchise Disclosure Document (FDD) is lengthy (300–700 pp +) and detailed (see Franchise Disclosure Document, above), and generally requires audited financial statements from the franchisor in a particular format, except in some circumstances, such as where a franchisor is new. It must include such data as the names, addresses and telephone numbers of the franchisees in the licensed territory (who may be contacted and consulted before negotiations), estimate of total franchise revenues and franchisor profitability.

Individual states may require the FDD to contain their own specific requirements, but the requirements in state disclosure documents must be in compliance with the federal rule that governs federal regulatory policy. There is no private right of action of action under the FTC rule for franchisor violation of the rule, but fifteen or more of the states have passed statutes that provide this right of action to franchisees when fraud can be proven under these special statutes. The majority of franchisors have inserted mandatory arbitration clauses into their agreements with their franchisees, some of which the U.S. Supreme Court has dealt with.

In response to the implementation of California Assembly Bill 5 (2019) which limits the use of classifying workers as independent contractors rather than employees in California, the United States Court of Appeals for the Ninth Circuit reinstated its decision in Vazquez v. Jan-Pro which impacts California franchise law and California independent contractor law by making it unclear that if a franchisor licenses its trademark to a franchisee, whether the franchisor incurs the liabilities of an employer for a franchisee's employees.

There is no federal registry of franchises or any federal filing requirements for information. States are the primary collectors of data on franchising companies and enforce laws and regulations regarding their presence and their spread in their jurisdictions.

Where the franchisor has many partners, the agreement may take the shape of a business format franchise – an agreement that is identical for all franchisees.

Social franchises
In recent years, the idea of franchising has been picked up by the social enterprise sector, which hopes to simplify and expedite the process of setting up new businesses. A number of business ideas, such as soap making, wholefood retailing, aquarium maintenance, and hotel operation have been identified as suitable for adoption by social firms employing disabled and disadvantaged people.

The most successful examples are probably the Kringwinkel second-hand shops employing 5,000 people in Flanders, franchised by KOMOSIE, the CAP Markets, a steadily growing chain of 100 neighbourhood supermarkets in Germany. and the Hotel Tritone in Trieste, which inspired the Le Mat social franchise, now active in Italy and Sweden.

Social franchising also refers to a technique used by governments and aid donors to provide essential clinical health services in the developing world.

Social Franchise Enterprises objective is to achieve development goals by creating self sustainable activities by providing services and goods in un-served areas. They use the Franchise Model characteristics to deliver Capacity Building, Access to Market and Access to Credit/Finance.

Third-party logistics franchising
Third-party logistics has become an increasingly more popular franchise opportunity due to the quickly growing transportation industry and low cost franchising. In 2012, Inc. Magazine ranked three logistics and transportation companies in the top 100 fastest growing companies in the annual Inc. 5000 rankings.

Event franchising
Event franchising is the duplication of public events in other geographical areas, retaining the original brand (logo), mission, concept and format of the event. As in classic franchising, event franchising is built on precisely copying successful events. An example of event franchising is the World Economic Forum, also known as the Davos forum, which has regional event franchisees in China, Latin America, etc. Likewise, the alter-globalist World Social Forum has launched many national events. When The Music Stops is an example of an events franchise in the UK, in this case, running speed dating and singles events.

Home-based franchises
The franchising or duplication of another firm's successful home-based business model is referred to as a home-based franchise. Home-based franchises are becoming popular as they are considered to be an easy way to start a business as they may provide a low barrier for entry into entrepreneurship. It may cost little to start a home-based franchise, but experts say that "the work is no less hard."

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title: Franchising