Georges Gras

UNLIMITED LISTS OF QUADRATIC INTEGERS OF GIVEN NORM APPLICATION TO SOME ARITHMETIC PROPERTIES

Keywords: Class groups). 2023. hal-03705446v2 Mathematics Subject Classification. Primary 11R11, 11R27, 11R37 Real quadratic fields, Fundamental units, Norm equations, p-rationality, p-class numbers, PARI programs. 1

Unlimited lists of fundamental units of quadratic fields -Applications to some arithmetic properties

Georges Gras

 3.3.Remarks on the use of the F.O.P. algorithm 4. Application of the F.O.P. algorithm to norm equations 4.1. Main property of the trace map for units 4.2. Unlimited lists of fundamental units of norm s, s ∈ {-1, 1} 4.3. Unlimited lists of fundamental integers of norm sν, ν ≥ 2 5. Universality of the polynomials m sν 5.1. Mc Laughlin's polynomials 5.2. Application to finding units 6. Non p-rationality of quadratic fields 6.1. Recalls about p-rationality 6.2. Remarks about p-rationality and non-p-rationality 6.3. Families of local p-th power units -Computation of T K 6.4. Infiniteness of non p-rational real quadratic fields 7. Application to p-class groups of some imaginary cyclic fields 7.1. Imaginary quadratic fields with non-trivial 3-class group 7.2. Imaginary cyclic fields with non-trivial p-class group, p > 3 References

Introduction and main results

1.1. Definition of the "F.O.P. " algorithm. For the convenience of the reader, we give, at once, an outline of this process which has an interest especially under the use of PARI programs [P].

Definition 1.1. We call "First Occurrence Process" (F.O.P. ) the following algorithm, defined on a large interval [1, B] of integers. As t grows from t = 1 up to t = B, we compute some arithmetic invariant F (t); for instance, a pair of invariants described as a PARI list, as the following illustration with square-free integers M (t) and units η(t) of Q( M (t)): F(t) → L(t) = List M(t), η(t) , provided with a natural order on the pairs L(t), then put it in a PARI list LM: Listput(LM, vector (2, c, L[c])) →List ([L(1), L(2), . . . , L(t), . . . , L(B)])

=List([ [M (1), η(1)], . . . , [M(t), η(t)], . . . , [M(B), η(B)] ]);

after that, we apply the PARI instruction VM = vecsort(LM, 1, 8) which builds the list:

VM = List([L 1 , L 2 , . . . , L j , . . . , L N ]), N ≤ B,
such that L j = L(t j ) = [ [M(t j ), η(t j )] ] is the first occurrence (regarding the selected order, for instance that on the M's) of the invariant found by the algorithm and which removes the subsequent duplicate entries.

Removing the duplicate entries is the key of the principle since in general they are unbounded in number as B → ∞ and do not give the suitable information Since the length N of the list VM is unknown by nature, one must write LM as a vector and put instead: VM = vecsort(vector (B, c, LM[c]), 1, 8); thus, N = #VM makes sense and one can (for possible testing) select elements and components as X = VM[k] [2], etc. If N is not needed, then VM = vecsort(LM, 1, 8) works well.

For instance, the list LM of objects F (t) = (M (t), ε(t)), 1 ≤ t ≤ B = 10:

LM = List([ [5, ε 5 ], [2, ε 2 ], [5, ε ′ 5 ], [7, ε 7 ], [5, ε ′′ 5 ], [3, ε 3 ], [2, ε ′ 2 ], [5, ε ′′′ 5 ], [6, ε 6 ], [7, ε ′ 7 ] ]
), with the natural order on the first components M , leads to the list: [3, ε 3 ], [5, ε 5 ], [6, ε 6 ], [7, ε 7 ]]).

VM = List([[2, ε 2 ],
1.2. Quadratic integers. Let K =: Q( √ M ), M ∈ Z ≥2 square-free, be a real quadratic field and let Z K be its ring of integers.

Recall that M ≥ 2, square-free, is called the "Kummer radical" of K, contrary to any "radical" m = M r 2 giving the same field K.

There are two ways of writing for an element α ∈ Z K . The first one is to use the integral basis {1, √ M } (resp. 1,

1+ √ M 2
) when M ≡ 1 (mod 4) (resp. M ≡ 1 (mod 4)). The second one is to write α = 1 2 (u + v √ M ), in which case u, v ∈ Z are necessarily of same parity; but u, v may be odd only when M ≡ 1 (mod 4). We denote by T K/Q and N K/Q , or simply T and N, the trace and norm maps in K/Q, so that T(α) = u and N(α) = 1 4 (u 2 -M v 2 ) in the second writing for α. Then the norm equation in u, v ∈ Z (not necessarily with co-prime numbers u, v):

u 2 -M v 2 = 4sν, s ∈ {-1, 1}, ν ∈ Z ≥1 ,
for M square-free, has the property that u, v are necessarily of same parity and may be odd only when M ≡ 1 (mod 4); then:

z := 1 2 u + v √ M ∈ Z K , T(z) = u & N(z) = sν.
Finally, we will write quadratic integers α, with positive coefficients on the basis {1, √ M }; this defines a unique representative modulo the sign and the conjugation. Put:

Z + K := α = 1 2 u + v √ M , u, v ∈ Z ≥1 , u ≡ v (mod 2) .
Note that these α's are not in Z, nor in Z• √ M ; indeed, we have the trivial solutions N(q) = q 2 (α = q ∈ Z ≥1 , s = 1, ν = q 2 , M v 2 = 0, u = q) or N(v

√ M ) = -M v 2 , v ∈ Z ≥1 , s = -1, ν = -M v 2 ,
u = 0), which are not given by the F.O.P. algorithm for simplicity. These viewpoints will be more convenient for our purpose and these conventions will be implicit in all the sequel.

Since norm equations may have several solutions, we will use the following definition: Definition 1.2. Let M ∈ Z ≥2 be a square-free integer and let s ∈ {-1, 1}, ν ∈ Z ≥1 . We call fundamental solution (if there are any) of the norm equation u 1.3. Quadratic polynomial units. It is classical that the continued fraction expansion of √ m, for a positive square-free integer m, gives, under some limitations, the fundamental solution, in integers u, v ∈ Z ≥1 , of the norm equation u 2 -mv 2 = 4s, whence the fundamental unit

2 -M v 2 = 4sν, with u, v ∈ Z ≥1 , the corresponding integer α := 1 2 (u + v √ M ) ∈ Z + K of minimal trace u.
ε m := 1 2 (u + v √ m) of Q( √ m).
A similar context of "polynomial continued fraction expansion" does exist and gives polynomial solutions (u(t), v(t)), of u(t) 2 -m(t)v(t) 2 = 4s, for suitable m(t) ∈ Z[t] (see, e.g., [McL, McLZ, Nat, Ram, SaAb]). This gives the quadratic polynomial units E(t) := 1 2 u(t) + v(t) m(t) . We will base our study on the following polynomials m(t) that have interesting universal properties (a first use of this is due to Yokoi [Yo, Theorem 1]).

Definition 1.3. Consider the square-free polynomials m sν (t) = t 2 -4sν ∈ Z[t], where s ∈ {-1, 1}, ν ∈ Z ≥1 . The continued fraction expansion of √ t 2 -4sν leads to the integers A sν (t) := 1 2 t + √ t 2 -4sν
, of norm sν and trace t, in a quadratic extension of Q(t). When ν = 1, one obtains the units E s (t) := 1 2 t + √ t 2 -4s , of norm s and trace t.

The continued fraction expansion, with polynomials, gives the fundamental solution of the norm equation (cf. details in [McL]), but must not be confused with that using evaluations of the polynomials; for instance, for

t 0 = 7, m 1 (t 0 ) = 7 2 -4 = 45 is not square-free and E 1 (7) = 1 2 (7 + √ 45) = 1 2 (7 + 3 √ 5) is indeed the fundamental solution of u 2 -45v 2 = 4, but not the fundamental unit ε 5 of Q( √ 45) = Q( √ 5
), since one gets E 1 (7) = ε 6 5 . 1.4. Main algorithmic results. We will prove that the families of polynomials m sν (t) = t 2 -4sν, s ∈ {-1, 1}, ν ∈ Z ≥1 , are universal to find all square-free integers M for which there exists a privileged solution α ∈ Z + K to N(α) = sν; moreover, the solution obtained is the fundamental one, in the meaning of Definition 1.2 saying that α is of minimal trace t ≥ 1. This is obtained by means of an extremely simple algorithmic process (described § 1.1) and allows to get unbounded lists of quadratic fields, given by means of their Kummer radical, and having specific properties.

The typical results, admitting several variations, are given by the following excerpt of statements using quadratic polynomial expressions m(t) deduced from some m sν (t):

Theorem 1.4. Let B be an arbitrary large upper bound. As the integer t grows from 1 up to B, for each first occurrence of a square-free integer M ≥ 2, in the factorizations m(t) =: M r 2 , we have the following properties for Definition 1.2 (Theorem 4.6). b) Let p be an odd prime number and consider the following polynomials.

K := Q( √ M ): a) Consider the polynomials m(t) = t 2 -4sν, s ∈ {-1, 1}, ν ∈ Z ≥2 : (i) m(t) = t 2 -4s. The unit 1 2 (t + r √ M ) is the fundamental unit of norm s of K (Theorem 4.2). (ii) m(t) = t 2 -4sν. The integer A sν (t) = 1 2 (t + r √ M ) is the fundamental integer in Z + K of norm sν in the meaning of
(i) m(t) ∈ p 4 t 2 ± 1, p 4 t 2 ± 2, p 4 t 2 ± 4, 4p 4 t 2 ± 2, 9p 4 t 2 ± 6, 9p 4 t 2 ± 12, . . . . The field K is non p-rational apart from few explicit cases (Theorem 6.3). (ii) m(t) = 3 4 t 2 -4s. The field F 3,M := Q( √ -3M
) has a class number divisible by 3, except possibly when the unit 1 2 9t + r √ M is a third power of a unit (Theorem 7.1). Up to B = 10 5 , all the 3-class groups are non-trivial, apart from few explicit cases.

(iii) m(t) = p 4 t 2 -4s, p ≥ 5.

The imaginary cyclic extension

F p,M := Q (ζ p -ζ -1 p ) √ M , of degree p -1,
has a class number divisible by p, except possibly when the unit 1 2

p 2 t + r √ M ) is a p-th power of unit (Theorem 7.2).
For p = 5, the quartic cyclic field F 5,M is defined by the polynomial P = x 4 +5M x 2 +5M 2 and up to B = 500, all the 5-class groups are non-trivial, except for M = 29. Moreover, this principle gives lists of solutions by means of Kummer radicals (or discriminants) of a regularly increasing order of magnitude, these lists being unbounded as B → ∞. See, for instance Proposition 3.1 for lists of Kummer radicals M , then Section 2 for lists of arithmetic invariants (class groups, p-ramified torsion groups, logarithmic class groups of K), and Theorems 6.4, 6.5, giving unlimited lists of units, local (but non global) pth powers, whence lists of non-p-rational quadratic fields.

All the lists have, at least, O(B) distinct elements, but most often B -o(B), and even B distinct elements in some situations.

So, we intend to analyze these results in a computational point of view by means of a new strategy to obtain arbitrary large list of fundamental units, or of other quadratic integers, even when radicals m sν (t) =: M (t)r(t) 2 , t ∈ Z ≥1 , are not square-free (i.e., r(t) > 1). By comparison, it is well known that many polynomials, in the literature, give subfamilies of integers (especially fundamental units) found by means of the m sν 's with assuming that the radical m sν (t) are square-free.

Remark 1.5. It is accepted and often proven that the integers t 2 -4sν are square-free with a non-zero density and an uniform repartition (see, e.g., [FrIw], [Rud]); so an easy heuristic is that the last M = M B of the list VM is equivalent to B 2 . This generalizes to the F.O.P. algorithm applied to polynomials of the form

a n t n + a n-1 t n-1 + • • • + a 0 , n ≥ 1, a n ∈ Z ≥1 , and gives the equivalent M B ∼ a n B n as B → ∞.
The main fact is that the F.O.P. algorithm will give fundamental solutions of norm equations u 2 -M v 2 = 4sν (see Section 4), whatever the order of magnitude of r; for small values of M , r may be large, even if r(t) tends to 1 as M (t) tends to its maximal value, equivalent to B 2 , as t → ∞. Otherwise, without the F.O.P. principle, one must assume m sν (t) square-free in the applications, as it is often done in the literature.

2.

First examples of application of the F.O.P. algorithm 2.1. Kummer radicals and discriminants given by m s (t). Recall that, for t ∈ Z ≥1 , we put m s (t) = M (t)r(t) 2 , M (t) square-free.

2.1.1. Kummer radicals. The following program gives, as t grows from 1 up to B, the Kummer radical M and the integer r obtained from the factorizations of m ′ 1 (t) = t 2 -1, under the form M r 2 ; then we put them in a list LM and the F.O.P. algorithm gives the pairs C = core(mt, 1) = [M, r], in the increasing order of the radicals M and removes the duplicate entries:

MAIN PROGRAM GIVING KUMMER RADICALS {B=10^6;LM=List;for(t=1,B,mt=t^2-1;C=core(mt,1);L=List(C); listput(LM,vector (2,c,L[c]))); M=vecsort(vector(B,c,LM[c]),1,8); print (M);print("#M = ",#M)} [M,r]= [0,1], [2,2], [3,1], [5,[START_REF]The case b) of Theorem 6.3 gives an analogous program and will be also illustrated in the Section 7 about p-class groups, especially for the case p = 3. The results are similar and give, in almost cases, non[END_REF], [6,2], [7,3], [10,6], [START_REF]Ln=List;LM=List; for(t=1[END_REF]3], [13,180], [14,[START_REF]The case b) of Theorem 6.3 gives an analogous program and will be also illustrated in the Section 7 about p-class groups, especially for the case p = 3. The results are similar and give, in almost cases, non[END_REF], [15,1], [17,8], [19,39], [START_REF]+ 5 * M * x 2 + 5 * M 2 , still giving a particular faster program than the forthcoming one, valuable for any p ≥ 3: LISTS OF 5-CLASS GROUPS OF QUARTIC FIELDS {p=5;B=100;s=-1;Lp=List;Lh=List;p2=p^2;p4=p^4;for([END_REF]12], [22,42], [23,5], [26,10], [29,1820], [30,2], [31,273], [33,[START_REF]The case b) of Theorem 6.3 gives an analogous program and will be also illustrated in the Section 7 about p-class groups, especially for the case p = 3. The results are similar and give, in almost cases, non[END_REF], [34,6], [35,1], [37,12], [38,6], [39,[START_REF]The case b) of Theorem 6.3 gives an analogous program and will be also illustrated in the Section 7 about p-class groups, especially for the case p = 3. The results are similar and give, in almost cases, non[END_REF], [41,320], [42,2], [43,531], [46,3588], [47,7], [51,7], [53,9100], [55,12], [57,20] {B=10^6;LD=List;for(t=1,[START_REF]T = prime(t) (traces are prime), s = -1 (B = 10 4[END_REF])]); listput(LD,vector (1,c,L[c])); L=List([quaddisc(core(t^2+1))]); listput(LD,vector (1,c,L[c])));D=vecsort(vector (2*B,c,LD[c]),1,8); print(D);print("#D = ",#D)} [D]= [[0], [5] This possibility is valid for all programs of the paper; we will classify the Kummer radicals, instead of discriminants, because radicals are more related to norm equations, but any kind of output can be done easily.

2.2. Application to minimal class numbers. One may use this classification of Kummer radicals and compute orders h of some invariants, then apply the F.O.P. principle, with the instruction VM = vecsort(vector(B, c, LM[c]), 2, 8) to the outputs [M, h], to get successive possible class numbers h in ascending order (we use here m 1 (t) = t 2 -4):

2.4. Application to minimal orders of logarithmic class groups. For the definition of the logarithmic class group T p governing Greenberg's conjecture [START_REF] Greenberg | On the Iwasawa invariants of totally real number fields[END_REF], see [START_REF] Jaulent | Classes logarithmiques des corps de nombres[END_REF]Jau4], and for its computation, see [BeJa] which gives the structure as abelian group. The following program, for p = 3, gives the results by ascending orders (all the structures are cyclic in this interval):

MAIN PROGRAM GIVING SUCCESSIVE CLASSLOG NUMBERS {B=10^5;LM=List;for(t=3,B,M1=core(t^2-4);M2=core(t^2+4); K1=bnfinit(x^2-M1);Clog= bnflog(K1,3) [1];C=1;for(j=1,#Clog, C=C*Clog[j]);L=List([M1,Clog,C]);listput (LM,vector(3,c,L[c])); K2=bnfinit(x^2-M2);Clog= bnflog(K2,3) [1];C=1;for(j=1,#Clog, C=C*Clog[j]);L=List([M2,Clog,C]);listput (LM,vector(3,c,L[c]))); [5,[] ,1],[257,[3],3],[2917,[9],9],[26245,[27],27],[577601,[81],81],[236197,[243],243], [19131877,[729],729], [172186885,[2187],2187], [1549681957,[6561],6561]] #VM = 9

VM=vecsort(vector(2*(B-2),c,LM[c]),3,8); print(VM);print("#VM = ",#VM)} [M,Clog,#Clog]= [
3. Units E s (t) vs fundamental units ε M(t)

3.1. Polynomials m s (t) = t 2 -4s and units E s (t). This subsection deals with the case ν = 1 about the search of quadratic units (see also [Yo,Theorem 1]). The polynomials

m s (t) ∈ Z[t] define, for t ∈ Z ≥1 , the parametrized units E s (t) = 1 2 (t + √ t 2 -4s) of norm s in K := Q( √ M )
, where M is the maximal square-free divisor of t 2 -4s. But M is unpredictable and gives rise to the following discussion depending on the norm

S := N(ε M ) of the fundamental unit ε M =: 1 2 (a + b √ M ) of K and of the integral basis of Z K : (i) If s = 1, E 1 (t) = 1 2 (t + √ t 2 -4) is of norm 1; so, if S = 1, then E 1 (t) ∈ ε M , but if S = -1, necessarily E 1 (t) ∈ ε 2 M . If s = -1, E -1 (t) = 1 2 (t + √ t 2 + 4) is of norm -1; so, necessarily the Kummer radical M is such that S = -1. (ii) If t is odd, E s (t) is written with half-integer coefficients, t 2 -4s ≡ 1 (mod 4), giving M ≡ 1 (mod 4) and Z K = Z 1+ √ M 2
; so ε M can not be with integer coefficients (a and b are necessarily odd).

If t is even, M may be arbitrary as well as ε M .

We can summarize these constraints by means of the following Table :   (3.1)

t 2 -4s S = N(ε M ) Es(t) ∈ ε M = 1 2 (a + b √ M ) t 2 -4, t even 1 (resp. -1) ε M (resp. ε 2 M ) a, b odd or even t 2 -4, t odd 1 (resp. -1) ε M (resp. ε 2 M ) a, b odd t 2 + 4, t even -1 ε M a, b odd or even t 2 + 4, t odd -1 ε M a, b odd
Recall that the F.O.P. algorithm consists, after choosing the upper bound B, in establishing the list of first occurrences, as t increases from 1 up to B, of any square-free integer M ≥ 2, in the factorization m s (t) = M (t)r(t) 2 (whence M = M (t 0 ) for some t 0 and M = M (t) for all t < t 0 ), and to consider the unit:

E s (t) := 1 2 t + t 2 -4s = 1 2 t + r(t) M (t) , of norm s. The F.O.P. is necessary since, if t 1 > t 0 gives the same Kummer radical M , E s (t 0 ) = ε n0 M and E s (t 1 ) = ε n1 M with n 1 > n 0 .
We shall prove (Theorem 4.2) that, under the F.O.P. algorithm, one always obtains the minimal possible power n ∈ {1, 2} in the writing E s (t) = ε n M , whence n = 2 if and only if s = 1 and S = -1, which means that E s (t) is always the fundamental unit of norm s.

The following result shows that any square-free integer M ≥ 2 may be obtained for B large enough.

Proposition 3.1. Consider the polynomial m 1 (t) = t 2 -4. For any square-free integer M ≥ 2, there exists t ≥ 1 such that m 1 (t) = M r 2 . Proof. The corresponding equation t 2 -4 = M r 2 becomes of the form t 2 -M r 2 = 4. Depending on the writing in Z[ √ M ] (M ≡ 2, 3 (mod 4)) or Z 1+ √ M 2 (M ≡ 1 (mod 4)), of the powers ε n M = 1 2 (t + r √ M ), n ≥ 1, of the fundamental unit ε M , this selects infinitely many t ∈ Z ≥1 . Remark 3.2. One may use, instead, the polynomial m ′ 1 (t) = t 2 -1 since for any funda- mental unit of the form ε M = 1 2 (a+b √ M ), a, b odd, then ε 3 M ∈ Z[ √ M ]
, but some radicals are then obtained with larger values of t; for instance, m 1 (5) = 21 and m ′ 1 (55) = 21 • 12 2 corresponding to 55 + 12

√ 21 = 1 2 (5 + √ 21) 2 .
Since for t = 2t ′ , t 2 + 4s = 4(t ′2 -s) gives the same Kummer radical as t ′2 -s, in some cases we shall use m ′ s (t) := t 2 -s and especially m ′ 1 (t) := t 2 -1 which is "universal" for giving all Kummer radicals.

With the polynomials m -1 (t) = t 2 + 4 or m ′ -1 (t) = t 2 + 1 a solution does exist if and only if N(ε M ) = -1 and one obtains odd powers of ε M .

Checking of the exponent

n in E s (t) = ε n M(t) .
The following program determines the expression of E s (t) as power of the fundamental unit of K; it will find that there is no counterexample to the relation

E s (t) ∈ {ε M(t) , ε 2 M(t)
}, depending on S, from Table (3.1); this will be proved later (Theorem 4.2). So these programs are only for verification, once for all, because they unnecessarily need much more execution time.

Since E s (t) is written in 1 2 Z[ √ M
] and ε M on the usual Z-basis of Z K denoted {1, w} by PARI (from the instruction quadunit), we write E s (t) on the PARI basis {1, quadgen(D)}, where D = quaddisc(M) is the discriminant.

One must specify B and s, the program takes into account the first value 2 + s of t since t = 1, 2 are not suitable when s = 1; then the test n > (3 + s)/2 allows the cases n = 1 or 2 when s = 1. The output of counterexamples is given by the (empty) list Vn: by the following ones (but any information can be put in L; the sole condition being to put M as first component): (LM,vector(3,c,L[c])), listput (LN,vector(3,c,Ln[c]))

L = List([M, n, t]), listput
or simply:

L = List([M, t]), listput(LM, vector(2, c, L[c])), listput(LN, vector(2, c, Ln[c]))
giving the parameter t whence the trace, then the whole integer of Q( √ M ); for instance for m -1 (t) = t 2 + 4 and the general program with outputs [M, n, t]:

(iii) When several polynomials m i (t), 1 ≤ i ≤ N , are considered together (to get more Kummer radicals solutions of the problem), there is in general commutativity of the two sequences in for(t = 1, B, for

(i = 1, N, mt = • • • )) and for(i = 1, N, for(t = 1, B, mt = • • • )).
But we will always use the first one.

Application of the F.O.P. algorithm to norm equations

We will speak of solving a norm equation in

K = Q( √ M ), for the search of integers α ∈ Z + K such that N(α) = sν, for s ∈ {-1, 1} and ν ∈ Z ≥1 given (i.e., α = 1 2 u + v √ M , u, v ∈ Z ≥1 ).
If the set of solutions is non-empty we will define the notion of fundamental solution; we will see that this definition is common to units (ν = 1) and non-units.

We explain, in Theorem 4.6, under what conditions such a fundamental solution for ν > 1 does exist, in which case it is necessarily unique and found by means of the F.O.P. , algorithm using m -1 (t) or m 1 (t) (depending in particular on S).

Note that the resulting PARI programs only use very elementary instructions and never the arithmetic ones defining K (as bnfinit, K.fu, bnfisintnorm, ...); whence the rapidity even for large upper bounds B.

4.1.

Main property of the trace map for units. In the case ν = 1, let S = N(ε M ); we will see that α defines the generator of the group of units of norm

s of Q( √ M ) when it exists (whence ε M if s = S or ε 2 M if S = -1 and s = 1). Theorem 4.1. Let M ≥ 2 be a square-free integer. Let ε = 1 2 (a + b √ M ) > 1 be a unit of K := Q( √ M ) (non-necessarily fundamental). Then T(ε n ) defines a strictly increasing sequence of integers for n ≥ 1. 1 Proof. Set ε = 1 2 (a -b √ M )
for the conjugate of ε and let s = εε = ±1 be the norm of ε;

then the trace of ε n is T n := ε n + ε n = ε n + s n
ε n . Thus, we have:

T n+1 T n = ε n+1 + s n+1 ε n+1 ε n + s n ε n = ε 2(n+1) + s n+1 ε n+1 × ε n ε 2n + s n = ε 2(n+1) + s n+1 ε 2n+1 + s n ε .
To prove the increasing, consider ε 2n+1 + s n ε and ε 2(n+1) + s n+1 , which are positive for all n since ε > 1; then:

(4.1) ∆ n (ε) := ε 2(n+1) + s n+1 -(ε 2n+1 + s n ε) = ε 2(n+1) -ε 2n+1 + s n+1 -s n ε = ε 2n+1 (ε -1) -s n (ε -s). (i) Case s = 1. Then ∆ n (ε) = (ε -1)(ε 2n+1 -1) is positive. (ii) Case s = -1. Then ∆ n (ε) = ε 2(n+1) -ε 2n+1 -(-1) n (ε + 1
). If n is odd, the result is obvious; so, it remains to look at the expression for n = 2k, k ≥ 1:

(4.2) ∆ 2k (ε) = ε 4k+2 -ε 4k+1 -ε -1. Let f (x) := x 4k+2 -x 4k+1 -x -1; then f ′ (x) = (4k + 2)x 4k+1 -(4k + 1)x 4k -1 and f ′′ (x) = (4k + 1)x 4k-1 [(4k + 2)x -4k] ≥ 0 for all x ≥ 1. Thus f ′ (x)
is increasing for all x ≥ 1 and since f ′ (1) = 0, f (x) is an increasing map for all x ≥ 1; so, for k ≥ 1 fixed, ∆ 2k (ε) is increasing regarding ε.

Since the smallest unit ε > 1 with positive coefficients is ε 0 := 1+ √ 5 2 ≈ 1.6180... we have to look, from (4.2), at the map F (z) := ε 4z+2 0 -ε 4z+1 0 -ε 0 -1, for z ≥ 1, to check if there exists an unfavorable value of k; so:

F ′ (z) := 4 log(ε 0 )ε 4z+2 0 -4 log(ε 0 )ε 4z+1 0 = 4 log(ε 0 )ε 4z+1 0 (ε 0 -1) > 0.
Since F (1) ≈ 4.2360 > 0, one gets ∆ n (ε) > 0 in the case s = -1, n even. 4.2. Unlimited lists of fundamental units of norm s, s ∈ {-1, 1}. We have the main following result.

Theorem 4.2. Let B ≫ 0 be given. Let m s (t) = t 2 -4s, s ∈ {-1, 1} fixed. Then, as t grows from 1 up to B, for each first occurrence of a square-free integer M ≥ 2 in the factorization m s (t) = M r 2 , the unit E s (t) = 1 2 (t + r √ M ) is the fundamental unit of norm s of Q( √ M ) (according to the Table (3.1) in § 3.1, we have E s (t) = ε M if s = -1 or if s = S = 1, then E s (t) = ε 2
M if s = 1 and S = -1). Proof. Let M 0 ≥ 2 be a given square-free integer. Consider the first occurrence t = t 0 giving m s (t 0 ) = M 0 r(t 0 ) 2 if it exists (existence always fulfilled for s = 1 by Proposition 3.1); whence

M 0 = M (t 0 ). Suppose that E s (t 0 ) = 1 2 t 0 + r(t 0 ) M (t 0 ) is not the fundamental unit of norm s, ε n0 M(t0) (n 0 ∈ {1, 2}) but a non-trivial power (ε n0 M(t0) ) n , n > 1. Put ε n0 M(t0) =: 1 2 (a+b M (t 0 )); from Table (3.1), n 0 ∈ {1, 2} is such that N(ε n0 M(t0) ) = s (recall that if s = 1 and S = -1, n 0 = 2, if S = s = 1, n 0 = 1; if s = -1, necessarily S = -1 and n 0 = 1,
otherwise there were no occurrence of M 0 for s = -1 and S = 1).

Then, Theorem 4.1 on the traces implies 0 < a < t 0 . We have:

a 2 -M (t 0 )b 2 = 4s and m s (a) = a 2 -4s =: M (a)r(a) 2 ; but these relations imply M (t 0 )b 2 = M (a)r(a) 2 , whence M (a) = M (t 0 ) = M 0 .
That is to say, the pair t 0 , M 0 ) compared to a, M (a) = M 0 ) , was not the first occurrence of M 0 (absurd).

Corollary 4.3. Let t ∈ Z ≥1 and let E 1 (t) = 1 2 t + √ t 2 -4 of norm 1. Then E 1 (t) is a square of a unit of norm -1, if and only if there exists t ′ ∈ Z ≥1 such that t = t ′2 + 2; thus E 1 (t) = 1 2 (t ′ + √ t ′2 + 4) 2 = (E -1 (t ′ )) 2 .
So, the F.O.P. algorithm, with m 1 (t) =:

M (t)r(t) 2 , gives the list of [M(t), t] for which 1 2 t+ √ t 2 -4 = ε 2 M (resp. ε M ) if t-2 = t ′2 (resp. if not).
S = -1. Then t ′2 + 4 = M r ′2 for t ′ minimal gives the fundamental unit ε M = 1 2 (t ′ + r ′ √ M ) and t 2 -4 = M r 2 , for t minimal, gives ε 2 M ; whence t = t ′2 + 2 and r = r ′ t ′ . For s = -1, hence m -1 (t) = t 2 + 4, t ∈ [1, B],
we know, from Theorem 4.2, that the F.O.P. algorithm gives always the fundamental unit

ε M of Q( √ M ) whatever its writing in Z[ √ M ] or in Z 1+ √ M 2
.

For s = 1 one obtains ε 2 M if and only if S = -1. So we can skip checking and use the following simpler program with larger upper bound B = 10 7 ; the outputs are the Kummer radicals [M] in the ascending order (specify B and s):

(27) = 1 2 (5+ √ 29) 2 = ε 2 29
. Some Kummer radicals giving units ε M of norm -1 do not appear up to B = 10 7 , e.g., M ∈ {241, 313, 337, 394, . . .}; but all the Kummer radicals M , such that S = -1, ultimately appear as B increases. So, as B → ∞, any unit is obtained, which suggests the existence of natural densities in the framework of the F.O.P. algorithm. More precisely, in the list LM (i.e., before using VM = vecsort(vector(B, c, LM[c]), 1, 8)), any Kummer radical M does appear in the list as many times as the trace of ε n M (n odd) is less than B, which gives for instance (B = 10 3 ): This fact with Corollaries 4.3,4.4 may suggest some analytic computations of densities (see a forthcoming paper [START_REF] Gras | New characterization of the norm of the fundamental unit of Q( √ M )[END_REF] for more details). For this purpose, we give an estimation of the gap #LM -#VM = B -#VM.

Theorem 4.5. Consider the F.O.P. algorithm for units, in the interval [1, B] and s ∈ {-1, 1}. Let ∆ be the gap between B and the number of results. Then, as B → ∞:

(i) For the polynomial m -1 (t) = t 2 + 4, ∆ ∼ B 1 3 , (ii) For the polynomial m 1 (t) = t 2 -4, ∆ ∼ B 1 2 ,
Proof. (i) In the list LM of Kummer radicals giving units of norm -1, we know, from Theorem 4.2, that one obtain first the fundamental unit ε 0 := ε M0 from the relation t 2 0 + 4 = M 0 r 2 0 , then its odd powers ε 2n+1

M0

for n ∈ [1, n max ] corresponding to some t n such that t 2 n + 4 = M 0 r 2 n and t n ≤ t max defined by the equivalence:

1 2 t max + r max M 0 ∼ 1 2 t 0 + r 0 M 0 2nmax+1
in an obvious meaning. Thus, the "maximal unit" is equivalent to B giving Of course there will be repetitions in the sum, but a more precise estimation is not necessary and we obtain an upper bound:

n max ∼ 1 2 log B log t 0 -1 .
∆ ∼ t∈[1,b] 1 2 log B log t -1 ∼ log b t∈[1,b] 1 log t ∼ log b • b log b ∼ b = B 1 3 .
(ii) In the case of norm 1, the list LM is relative to the fundamental units of norm 1 with all its powers (some are the squares of the fundamental units of norm -1); the reasoning is the same, replacing 1 3 by 1 2 . 4.3. Unlimited lists of fundamental integers of norm sν, ν ≥ 2. The F.O.P. algorithm always give lists of results, but contrary to units, some norms sν do not exist in a given field K; in other words, the F.O.P. only give suitable Kummer radicals since sν is given. Recall the well known: Theorem 4.6. Let s ∈ {-1, 1} and ν ∈ Z ≥2 be given.

(i) A fundamental solution of the norm equation u 2 -M v 2 = 4sν (Definition 1.2) does exist if and only if there exists an integer principal ideal a of absolute norm ν with a generator α ∈ Z + K whose norm is of sign s. Under the existence of a = (α), with N(α) = s ′ ν, a representative α ∈ Z + K , modulo ε M , does exist whatever s as soon as S = -1; if S = 1, a fundamental solution α ∈ Z + K does exist if and only if s ′ = s.

(ii) When the above conditions are fulfilled, the fundamental solution corresponding to the ideal a is unique (in the meaning that two generators of a in Z + K , having same trace, are equal) and found by the F.O.P. algorithm.

Proof. (i) If a = (α), of absolute norm ν, with α = 1 2 (u + v √ M ) ∈ Z + K , one obtains u 2 -M v 2 = 4sν for a suitable s ∈ {-1, 1} giving a solution with t = u; then m sν (t) = t 2 -4sν = M (t)r 2 , whence M = M (u) and r = v.
Reciprocally, assume that the corresponding equation (in unknowns t ≥ 1, s = ±1) t 2 -4sν = M r 2 , M ≥ 2 square-free, has a solution, whence t 2 -M r 2 = 4sν. Set α := 1 2 (t + r √ M ) ∈ Z + K ; then one obtains the principal ideal a = (α)Z K of absolute norm ν.

(ii) Assume that α, β are two generators of a in Z + K with common trace t ≥ 1. Put

β = α • ε n M , n ∈ Z, n = 0. Then: T(β) = α • ε n M + α σ • ε nσ M = α 2 • ε 2n M + sS n ν α • ε n M , T(α) = α 2 + s ν α ; thus T(β) = T(α) is equivalent to α 2 • ε 2n M + sS n ν = α 2 • ε n M + s νε n M , whence to: α 2 • ε n M (ε n M -1) = (ε n M -S n )s ν.
The case

S n = -1 is not possible since N(β) = N(α) = N(ε n M ) = S n ; so, S n = 1, in which case, one gets α 2 • ε n M = s ν = s α 1+σ , thus α σ = α • ε n M and β = α σ , but in that case, β / ∈ Z + K (absurd)
. Whence the unicity. Remark 4.7. Consider the above case where α and β are two generators of a in Z + K with common trace t ≥ 1 and norm sν. Thus, we have seen that β = α σ = α • ε n M . The ideal a = (α) is then invariant by G := Gal(K/Q), so it is of the form a = (q) × p|D p ep , where q ∈ Z, D is the discriminant of K, p 2 = pZ K and e p ∈ {0, 1}. In other words, we have to determine the principal ideals, products of distinct ramified prime ideals. This is done in details in [Gra10, § § 2.1, 2.2] For instance, let M = 15 and sν = -6. One has the fundamental solution α = 3 + √ 15 of norm -6, with the trace t = 6; then

α σ = 3 - √ 15 = α • (-4 + √ 15) = α • (-ε -1 M )
; similarly, for sν = 10, one has the fundamental solution α = 5 + √ 15 of norm 10, with trace t = 10 and the relation

α σ = 5 - √ 15 = α • (4 - √ 15) = α • (ε σ M )
. These fundamental solutions are indeed given by the F.O.P. algorithm by means of the data [M, t]: [7,2], [10,[START_REF]The case b) of Theorem 6.3 gives an analogous program and will be also illustrated in the Section 7 about p-class groups, especially for the case p = 3. The results are similar and give, in almost cases, non[END_REF], [15,6] [-31,3], [-15,5], [-6,4], [-1,2], [1,7], [6,8], [10,20],[15,10],... Depending on the choice of the polynomials m -1 (t) or m 1 (t), consider for instance, the F.O.P. algorithm applied to M = 13 (for which S = -1), ν = 3, gives with m -1 (t) the solution [M = 13, t = 1] whence α = 1 2 (1 + √ 13) of norm -3; with m 1 (t) it gives [M = 13, t = 5], α = 1 2 (5 + √ 13) of norm 3; the traces 1 and 5 are minimal for each case. We then compute that 1 2 (5 + √ 13) = 1 2 (1 -√ 13)(-ε 13 ). But with M = 7 (for which S = 1), the F.O.P. algorithm with m -1 (t) and ν = 3 gives [M = 7, t = 4] but nothing with m 1 (t).

s.Nu=-6 [M,t]= [1,1],[6,24],
,... s.Nu=10 [M,t]= [-39,1],
Remark 4.8. A possible case is when there exist several principal integer ideals a of absolute norm νZ (for instance when ν = q 1 q 2 is the product of two distinct primes and if there exist two prime ideals q 1 , q 2 , of degree 1, over q 1 , q 2 , respectively, such that a := q 1 q 2 and a ′ := q 1 q σ 2 are principal). Let a =: (α) and a ′ =: (α ′ ) of absolute norm ν. We can assume that, in each set of generators, α and α ′ have minimal trace u and u ′ , and necessarily we have, for instance, u ′ > u; since the ideals a are finite in number, there exists an "absolute" minimal trace u defining the unique fundamental solution which is that found by the suitable F.O.P. algorithm.

For instance, let s = -1, ν = 15; the F.O.P. algorithm gives the solution [19,[START_REF]The case b) of Theorem 6.3 gives an analogous program and will be also illustrated in the Section 7 about p-class groups, especially for the case p = 3. The results are similar and give, in almost cases, non[END_REF], whence

α = 2 + √ 19 of norm -15. In K = Q( √ 19) we have prime ideals q 3 = (4 + √ 19) | 3, q 5 = (9 + 2 √ 19) | 5.
Then we obtain the fundamental solution with a = q σ 3 q 5 , while q 3 q 5 = (74 + 17 √ 19). The fundamental unit is ε M = 170 + 39 √ 19 of norm S = 1 and one computes some products ±αε n M giving a minimal trace with n = -1 and the non-fundamental solution 17 + 4 √ 19.

If ν = q|ν q nq , where q denotes distinct prime numbers, there exist integer ideals a of absolute norm νZ if and only if, for each inert q | ν then n q is even. In the F.O.P. algorithm this will select particular Kummer radicals M for which each q | ν, such that n q is odd, ramifies or splits in

K = Q( √ M ); this is equivalent to q | D (the discriminant of K = Q( √ M
)) or to ρ q := M q = 1 in terms of quadratic residue symbols; if so, we then have ideal solutions N(a) = νZ.

Let's write, with obvious notations a = q,q =0 q nq q, ρq=-1 q 2n ′ q q, ρq =1 q n ′ q q n ′′ q σ . Then the equation becomes N(a ′ ) = ν ′ Z for another integral ideal a ′ and another ν ′ | ν, where a ′ is an integer ideal "without any rational integer factor". Thus, N(α ′ ) = sν ′ is equivalent to a ′ = α ′ Z K . This depends on relations in the class group of K and gives obstructions for some Kummer radicals M . Once a solution a ′ principal exists (non unique) we can apply Theorem 4.6. For instance, [85,5] illustrates Theorem 4.6 with the solution α = 1 2 (5 + √ 85) of norm -15, with (α)Z K = q 3 q 5 , where 3 splits in K and 5 is ramified; one verifies that the ideals q 3 and q 5 are non-principal, but their product is of course principal. For this, one obtains the following PARI/GP verifications: [2,[2], [[3,1;0,1]]] idealfactor(k,3)=[ [3,[0,2] [3,[2,2] [3,[2,2]~,1,1,[0,-1]~], [5,[1,2]

k=bnfinit(x^2-85) k.clgp=
~,1,1,[-1,-1]~]1],[[3,[2,2]~,1,1,[0,-1]~]1] idealfactor(k,5)=[[5,[1,2]~,2,1,[1,2]~]2] bnfisprincipal(k,
~,1,1,[0,-1]~])=[[1]~,[1,0]~] bnfisprincipal(k,[5,[1,2]~,2,1,[1,2]~])=[[1]~,[1,1/3]~] A=idealmul(k,
~,2,1,[1,2]~]) bnfisprincipal(k,A)=[[0]~,[2,-1]~] nfbasis(x^2-85)=[1,1/2*x-1/2]
The data [[0], [2, -1]] gives the principality with generator [2, -1]

denoting (because of the integral basis {1, 1 2 x -1 2 } used by PARI), 2 -1 2 √ 85 -1 2 = 1 2 (5 - √ 85) = α σ . (iv) s = -1, ν = 9 × 25.
The case [37,5] may be interpreted as follows:

m -1 (5) = 5 2 + 4 • 9 • 25 = 5 2 • 37, whence A -1 (5) = 1 2 (5 + 5 √ 37) = 5 • 1 2 (1 + √ 37) =: 5B, where B := 1 2 (1 + √ 37
) is of norm -9 and 5 is indeed inert in K. Thus 1 2 (1 + √ 37)Z K is the square of a prime ideal q 3 over 3. The field K is principal and we compute that

q 3 = 1 2 (5 ± √ 37)Z K , q 2 3 = 1 2 (31 ± 5 √ 37)Z K . So, BZ K = 1 2 (1 + √ 37)Z K = 1 2 (31 + 5 √ 37)Z K or 1 2 (31 -5 √ 37)Z K .
We have ε 37 = 6 + √ 37 and we obtain that

B = 1 2 (31 -5 √ 37) • ε 37 , showing that α = 5 • 1 2 (1 + √ 37)
is the fundamental solution of the equation N(α) = 3 2 • 5 2 with minimal trace 5.

For larger integers ν, fundamental solutions are obtained easily, as shown by the following example with the prime ν = 1009: (v) s = -1, ν = 1009.

We have not dropped the negative radicals meaning, for instance with M = -839, that a solution of the norm equation does exist in Q( √ -839) with α = 1 2 (1 + √ -839), or with M = -14 giving α = 14 + √ -14.

Universality of the polynomials m sν

Let's begin with the following obvious result making a link with polynomials m sν .

Lemma 5.1. Let M ≥ 2 be a square-free integer and

K = Q( √ M ); then, any α ∈ Z + K is characterized by its trace a ∈ Z and its norm sν, s ∈ {-1, 1}, ν ∈ Z ≥1 ; from these data, α = 1 2 (a + b √ M ) where b is given by m sν (a) =: M b 2 .
Proof. From the equation α 2 -aα+sν = 0, we get α = 1 2 (a+ √ a 2 -4sν), where necessarily a 2 -4sν =: M b 2 (unicity of the Kummer radical) giving b > 0 from the knowledge of a and sν.

Mc Laughlin's polynomials.

Consider some polynomials that one finds in the literature; for instance that of Mc Laughlin [McL] obtained from "polynomial continued fraction expansion", giving formal units, and defined as follows.

Let m ≥ 2 be a given square-free integer and let 2,3,6}). For such m, u, v, each of the data below leads to the fundamental polynomial solution of the norm equation U (t) 2 -m(t)V (t) 2 = 1 (see [START_REF] Laughlin | Polynomial Solutions of Pell's Equation and Fundamental Units in Real Quadratic Fields[END_REF]), giving the parametrized units

E m = u + v √ m, u, v ∈ Z ≥1 , be the fundamental solution of the norm equation (or Pell-Fermat equation) u 2 -mv 2 = 1 (thus, E m = ε n0 m , n 0 ∈ {1,
E M(t) = U (t) + V (t)r M (t), of norm 1 of Q( M (t))
, where m(t) =: M (t)r(t) 2 , M (t) square-free.

The five polynomials m(t) are:

                                                   mcl 1 (t) = v 2 t 2 + 2ut + m, U (t) = v 2 t + u, V (t) = v; mcl 2 (t) = (u -1) 2 v 2 t 2 + 2t + m, U (t) = (u -1) v 4 t 2 + 2v 2 t + u, V (t) = v 3 t + v; mcl 3 (t) = (u + 1) 2 v 2 t 2 + 2t + m, U (t) = (u + 1) v 4 t 2 + 2v 2 t + u, V (t) = v 3 t + v; mcl 4 (t) = (u + 1) 2 v 2 t 2 + 2(u 2 -1)t + m, U (t) = (u + 1) 2 u -1 v 4 t 2 + 2(u + 1)v 2 t + u, V (t) = u + 1 u -1 v 3 t + v; mcl 5 (t) = (u -1) 2 v 6 t 4 + 4v 4 t 3 + 6v 2 t 2 + 2(u -1)(2u -1)t + m, U (t) = (u -1) v 6 t 3 + 3v 4 t 2 + 3v 2 t + u, V (t) = v 3 t + v.
Note that for mcl 1 (t) one may also use a unit

E m = u + v √ m of norm -1 since U (t) 2 -mcl 1 (t)V (t) 2 = u 2 -mv 2
, which is not possible for the other polynomials.

We may enlarge the previous list with cases where the coefficients of E m may be halfintegers defining more general units (as ε 5 , ε 13 of norm -1 in the case of mcl 1 (t), then as ε 21 of norm 1 for the other mcl(t)). This will give E m = ε m or ε 2 m . So we have the following transformation of the mcl(t), U (t), V (t), that we explain with mcl 1 (t). The polynomial mcl 1 (t) fulfills the condition

U (t) 2 -mcl 1 (t)V (t) 2 = u 2 -mv 2 , which is the norm of E m = u + v √ m; so we can use any square-free integer m ≡ 1 (mod 4) such that E m = 1 2 u + v √ m , u, v ∈ Z ≥1
odd, and we obtain the formal unit

E M(t) = 1 2 U (t) + V (t) mcl 1 (t)
under the condition t even to get U (t), V (t) ∈ Z ≥1 . This gives the polynomials mcl 6 (t) = v 2 t 2 +2ut+m and the coefficients

U (t) = 1 2 (v 2 t+u), V (t) = 1 2 v of a new unit, with mcl 6 (t) = M (t)r(t) 2
, for all t ≥ 0,. For the other mcl(t) one applies the maps t → 2t, t → 4t, depending on the degrees; so we obtain the following list, where the resulting unit is E M(t) = U (t) + V (t) m(t), of norm ±1, under the conditions m ≡ 1 (mod 4) and

ε m = 1 2 (u + v √ m), u, v odd:                                                              mcl 6 (t) = v 2 t 2 + 2ut + m, U (t) = 1 2 (v 2 t + u), V (t) = 1 2 v; mcl 7 (t) = (u -2) 2 v 2 t 2 + 2t + m, U (t) = 1 2 (u -2)(v 4 t 2 + 2v 2 t) + u , V (t) = 1 2 v 3 t + v ; mcl 8 (t) = (u + 2) 2 v 2 t 2 + 2t + m, U (t) = 1 2 (u + 2)(v 4 t 2 + 2v 2 t) + u , V (t) = 1 2 v 3 t + v ; mcl 9 (t) = (u + 2) 2 v 2 t 2 + 2(u 2 -4)t + m, U (t) = 1 2 (u + 2) 2 u -2 v 4 t 2 + 2(u + 2)v 2 t + u , V (t) = 1 2 u + 2 u -2 v 3 t + v ; mcl 10 (t) = (u -2) 2 v 6 t 4 + 4v 4 t 3 + 6v 2 t 2 + 4(u -2)(u -1)t + m, U (t) = 1 2 (u -2)(v 6 t 3 + 3v 4 t 2 + 3v 2 t) + u , V (t) = 1 2 v 3 t + v .
5.2. Application to finding units. In fact, these numerous families of parametrized units are nothing but the units E s (T ) = 1 2 (T + √ T 2 -4s) when the parameter T = U (t) is a given polynomial expression. This explain that the properties of the units E s (T ) are similar to that of the two universal units E s (t), for t ∈ Z ≥1 , but, a priori, the F.O.P. algorithm does not give fundamental units when T (t) is not a degree 1 monic polynomial; nevertheless it seems that the algorithm gives most often fundamental units, at least for all t ≫ 0. We give the following example, using for instance the Mc Laughlin polynomial mcl 10 (t) with m = 301, u = 22745, v = 1311, corresponding to, ε m = 1 2 (22745 + 1311 √ 301) of norm 1 (program of Section 3); this will give enormous units E M(t) =: ε n M(t) . The output is of the form [M(t), r(t), n]. Then there is no exception to E M(t) = ε M(t) (i.e., n = 1); moreover, one sees many cases of non-square-free integers mcl 10 (t):

Mc LAUGHLIN UNITS {B=10^3;LN=List;LM=List;u=22745;v=1311;for(t=1,B, mt=(u-2)^2*(v^6*t^4+4*v^4*t^3+6*v^2*t^2)+4*(u-2)*(u-1)*t+301; ut=1/2*((u-2)*(v^6*t^3+3*v^4*t^2+3*v^2*t)+u);vt=1/2*(v^3*t+v); C=core(mt,1);M=C [1];r=C [2];D=quaddisc (M);w=quadgen(D); Y=quadunit(D);res=Mod (M,4); if(res!=1,Z=ut+r*vt*w);if(res==1,Z=ut-r*vt+2*r*vt*w); z=1;n=0;while(Z!=z,z=z*Y;n=n+1);L=List ([M,r,n]); listput (LM,vector(3,c,L[c]))); VM=vecsort(vector(B,c,LM[c]),1,8); print(VM);print("#VM = ",#VM); for(k=1,#VM,n=VM[k] [3];if(n>1,Ln=VM[k]; listput (LN,vector(3,c,Ln[c]))));Vn=vecsort(LN,1,8); print("exceptional powers:",Vn)} [M,r,n] = [656527122296918386395032242,2,1],[1594671238615711306590405613,63245,1],[6538031892707128354912512481,1400,1],[8374054846220987469202089646,14,1] Using the Remark 1.5, with B = 10 3 and m(t) of degree 4, with leading coefficient:

a 4 B 4 = (22745 -2) 2 • 1311 6 • 10 12 = 2626102377422775499879732689000000000000,
one gets: log(2626102383534535069268098426753041168301)/ log(a 4 B 4 ) ≈ 1.000000000025... 6. Non p-rationality of quadratic fields 6.1. Recalls about p-rationality. Let p ≥ 2 be a prime number. The definition of p-rationality of a number field lies in the framework of abelian p-ramification theory. The references we give in this article are limited to cover the subject and concern essentially recent papers; so the reader may look at the historical of the abelian p-ramification theory that we have given in [START_REF] Gras | Practice of the Incomplete p-Ramification Over a Number Field -History of Abelian p-Ramification[END_REF]Appendix], for accurate attributions, from Šafarevič's pioneering results, about the numerous approaches (class field theory, Galois cohomology, pro-p-group theory, infinitesimal theory); then use its references concerning developments of this theory (from our Crelle's papers 1982-1983, Jaulent's infinitesimals [START_REF] Jaulent | S-classes infinitésimales d'un corps de nombres algébriques[END_REF] (1984), Jaulent's thesis [START_REF] Jaulent | L'arithmétique des ℓ-extensions Thèse de doctorat d'Etat[END_REF] (1986), Nguyen Quang Do's article [Ng] (1986), Movahhedi's thesis [Mov] (1988) and subsequent papers); all prerequisites and developments are available in our book [Gra1] (2005). Definition 6.1. A number field K is said to be p-rational if K fulfills the Leopoldt conjecture at p and if the torsion group T K of the Galois group of the maximal abelian p-ramified (i.e., unramified outside p and ∞) pro-p-extension of K is trivial.

We will use the fact that, for totally real fields K, we have the formula:

(6.1) #T K = #C ′ K • #R K • #W K , where C ′
K is a subgroup of the p-class group C K and where W K depends on local and global p-roots of unity; for

K = Q( √ M ) and p > 2, C ′ K = C K and W K = 1 except if p = 3 and M ≡ -3 (mod 9), in which case W K ≃ Z/3Z. For p = 2, C ′ K = C K except if K( √ 2)/K is unramified (i.e., if M = 2M 1 , M 1 ≡ 1 (mod 4))
. Then R K is the "normalized p-adic regulator" of K (general definition for any number field in [START_REF] Gras | The p-adic Kummer-Leopoldt Constant: Normalized p-adic Regulator[END_REF]Proposition 5.2]). For

K = Q( √ M ) and p = 2, #R K ∼ 1 p log p (ε M ); for p = 2, #R K ∼ 1 2 d log 2 (ε M )
, where d ∈ {1, 2} is the number of prime ideals above 2.

So #T K is divisible by the order of R K , which gives a sufficient condition for the nonp-rationality of K. Since C K = W K = 1 for p ≫ 0, the p-rationality only depends on R K in almost all cases. Proposition 6. 2. ([Gra6,Proposition 5.1]) Let K = Q( √ m) be a real quadratic field of fundamental unit ε m . Let p > 2 be a prime number with residue degree f ∈ {1, 2}.

(i) For p ≥ 3 unramified in K, v p (#R K ) = v p (ε p f -1 m -1) -1. (ii) For p > 3 ramified in K, v p (#R K ) = 1 2 (v p (ε p-1 -1) -1)
, where p 2 = (p).

(iii) For p = 3 ramified in

K, v 3 (#R K ) = 1 2 (v p (ε 6 -1) -2 -δ)
, where p 2 = (3) and δ = 1 (resp. δ = 3) if m ≡ -3 (mod 9) (resp. m ≡ -3 (mod 9)).

A sufficient condition for the non-triviality of R K that encompasses all cases (since the decomposition of p in Q( M (t)) is unpredictable in the F.O.P. algorithm) is log p (ε m ) ≡ 0 (mod p 2 ); this implies that ε m is a local pth power at p. It suffices to force the parameter t to be such that a suitable prime-to-p power of E s (t) = 1 2 t + r(t) M (t) is congruent to 1 modulo p 2 . So, exceptions may arrive only when E s (t) is a global pth power. 6.2. Remarks about p-rationality and non-p-rationality. In some sense, the prationality of K comes down to saying that the p-arithmetic of K is as simple as possible and that, on the contrary, the non p-rationality is the standard context, at least for some p for K fixed and very common when K varies in some families, for p fixed. a) In general, most papers intend to find p-rational fields, a main purpose being to prove the existence of families of p-rational quadratic fields (see, e.g., [START_REF] Assim | Half-integral weight modular forms and real quadratic p-rational fields[END_REF][START_REF] Benmerieme | Les corps multi-quadratiques p-rationnels[END_REF][START_REF] Boeckle | Wieferich Primes and a mod p Leopoldt Conjecture[END_REF]BeMo,[START_REF] Bouazzaoui | Fibonacci numbers and real quadratic p-rational fields[END_REF][START_REF] Barbulescu | Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's p-rationality conjecture[END_REF]By,[START_REF] Gras | Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques[END_REF][START_REF] Gras | Heuristics and conjectures in the direction of a p-adic Brauer-Siegel theorem[END_REF]Kop,[START_REF] Maire | Composantes isotypiques de pro-p-extensions de corps de nombres et p-rationalité[END_REF][START_REF] Maire | A note on p-rational fields and the abc-conjecture[END_REF][START_REF] Chattopadhyay | On the p-rationality of consecutive quadratic fields[END_REF]); for this there are three frameworks that may exist in general, but, to simplify, we restrict ourselves to real quadratic fields:

(i) The quadratic field K is fixed and it is conjectured that there exist only finitely many primes p > 2 for which K is non p-rational, which is equivalent to the existence of finitely many p for which 1 p log p (ε K ) ≡ 0 (mod p). (ii) The prime p > 2 is fixed and it is proved/conjectured that there exist infinitely many p-rational quadratic field K, which is equivalent to the existence of infinitely many K's for which the p-class group is trivial and such that 1 p log p (ε K ) is a p-adic unit; this aspect is more difficult because of the p-class group.

(iii) One constructs some families of fields K(p) indexed by p prime. These examples of quadratic fields often make use of Lemma 5.1 to get interesting radicals and units.

For instance we have considered in [START_REF] Gras | Heuristics and conjectures in the direction of a p-adic Brauer-Siegel theorem[END_REF]§ 5.3] (as many authors), the polynomials t 2 p 2ρ + s for p-adic properties of the unit E = t 2 p 2ρ + s + tp ρ t 2 p 2ρ + 2s of norm 1.

Taking "ρ = 1 2 , t = 1", one gets the unit E = p + s + p(p + 2s) considered in [Ben] where it is proved that for p > 3, the fields Q( p(p + 2)) are p-rational since the p-class group is trivial (for analytic reasons) and the unit p+1+ p(p + 2) is not a local p-power. Note that 4p(p + 2s) = m 1 (2p + 2s), since N(E) = 1 for all s.

Similarly, in [BeMo], is considered the bi-quadratic fields Q( p(p + 2), p(p -2)), containing the quadratic field Q( p 2 -4) giving the unit 1 2 (p + p 2 -4) still associated to m 1 (p); the p-rationality comes from the control of the p-class group since the p-adic regulators are obviously p-adic units.

Finally, in [Kop], is considered the tri-quadratic fields Q( p(p + 2), p(p -2), √ -1) which are proven to be p-rational for infinitely many primes p; but these fields are imaginary, so that one has to control the p-class group by means of non-trivial analytic arguments.

The p-rational fields allow many existence theorems and conjectures (as the Greenberg's conjecture [START_REF] Greenberg | Galois representation with open image[END_REF] on Galois representations with open images, yielding to many subsequent papers as [START_REF] Assim | Half-integral weight modular forms and real quadratic p-rational fields[END_REF][START_REF] Benmerieme | Les corps multi-quadratiques p-rationnels[END_REF]BeMo,[START_REF] Bouazzaoui | Fibonacci numbers and real quadratic p-rational fields[END_REF][START_REF] Barbulescu | Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's p-rationality conjecture[END_REF]GrJa,[START_REF] Jaulent | L'arithmétique des ℓ-extensions Thèse de doctorat d'Etat[END_REF]Kop]); they give results in the pro-p-group Galois theory [START_REF] Maire | Composantes isotypiques de pro-p-extensions de corps de nombres et p-rationalité[END_REF]. Algorithmic aspects of p-rationality may be found in [START_REF] Gras | On p-rationality of number fields[END_REF][START_REF] Gras | Practice of the Incomplete p-Ramification Over a Number Field -History of Abelian p-Ramification[END_REF][START_REF] Pitoun | Computing the torsion of the p-ramified module of a number field[END_REF] and in [BeJa] for the logarithmic class group having strong connexions with T K in connection with another Greenberg conjecture [Gre1] (Iwasawa's invariants λ = µ = 0 for totally real fields); for explicit characterizations in terms of pramification theory, see [Jau4, Gra8], Greenberg's conjecture being obvious when T K = 1.

b)

We observe with the following program that the polynomials: m s (p + 1) = (p + 1) 2 -4s and m s (2p + 2) = 4(p + 1) 2 -4s always give p-rational quadratic fields, apart from very rare exceptions (only four ones up to 10 6 ) due to the fact that the units E s (p + 1) = 1 2 p + 1 + (p + 1) 2 -4s and E s (2p + 2) = p + 1 + (p + 1) 2 -s may be a local p-power as studied in [START_REF] Gras | Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques[END_REF] in a probabilistic point of view (except in the case of E 1 (2p + 2) = 1 + p + p 2 + 2p ≡ 1 (mod p), with p 2 = (p), thus never local pth power):

{nu=8;L=List ([-4,-1,1,4]);for (j=1,4,d=L[j]; print("m(p)=(p+1)^2-(",d,")");forprime(p=3,10^6, M=core((p+1)^2-d);K=bnfinit(x^2-M); wh=valuation(K.no,p); Kmod=bnrinit(K,p^nu);CKmod=Kmod.cyc;val=0;d=#CKmod;for(k=1,; w=valuation (Cl,p);if(w>0,val=val+w));if(val>0, print("p=",p," M=",M," v_p(#(p-class group))=",wh, " v_p(#(p-torsion group) The case of p = 3, M = 15 does not come from the regulator, nor from the class group, but from the factor #W K = 3 since 15 ≡ -3 (mod 9); but this case must be considered as a trivial case of non-p-rationality. c) For real quadratic fields, the 2-rational fields are characterized via a specific genus theory and are exactly the subfields of the form Q( √ m) for m = 2, m = ℓ, m = 2ℓ, where ℓ is a prime number congruent to ±3 (mod 8) (see proof and history in [START_REF] Gras | On p-rationality of number fields[END_REF]Examples IV.3.5.1]). So we shall not consider the case p = 2 since the non-2-rational quadratic fields may be easily deduced, as well as fields with non-trivial 2-class group. d) Nevertheless, these torsion groups T K are "essentially" the Tate-Šafarevič groups (see their cohomological interpretations in [Ng]):

)=",val))))} m(p)=(p+1)^2+4, p=13 M=2
III 2 K := Ker H 2 (G K,Sp , F p ) → p∈Sp H 2 (G Kp , F p ) ,
where S p is the set of p-places of K, G K,Sp the Galois group of the maximal S p -ramified pro-p-extension of K and G Kp the local analogue over K p ; so their non-triviality has an important arithmetic meaning about the arithmetic complexity of the number fields (see for instance computational approach of this context in [START_REF] Gras | Tate-Shafarevich groups in the cyclotomic Z-extension and Weber's class number problem[END_REF] for the pro-cyclic extension of Q and the analysis of the Greenberg's conjecture [START_REF] Greenberg | On the Iwasawa invariants of totally real number fields[END_REF] in [START_REF] Gras | Algorithmic complexity of Greenberg's conjecture[END_REF]). When the set of places S does not contain S p , few things are known about G K,S ; see for instance Maire's survey [Mai] and its bibliography, then [Gra5, Section 3] for numerical computations.

In other words, the non-p-rationality (equivalent, for p > 2, to III 2 K = 0) is an obstruction to a local-global principle and is probably more mysterious than p-rationality. Indeed, in an unsophisticated context, it is the question of the number of primes p such that the Fermat quotient 2 p-1 -1 p is divisible by p, for which only two solutions are known; then non-p-rationality is the same problem applied to algebraic numbers, as units ε M ; this aspect is extensively developed in [START_REF] Gras | Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques[END_REF] for arbitrary Galois number fields).

6.3. Families of local p-th power units -Computation of T K . We shall force the non triviality of R K to obtain the non-p-rationality of K.

(T ) = 1 2 T + √ T 2 -4 = 1 δ ap 4 t 2 -δs+p 2 t a 2 p 4 t 2 -2δas
, of norm 1 and local pth power at p. The cases (a, δ) ∈ {(1, 1), (1,2), (2,1), (3,1), (3,2), (4, 1), (5,1), (5,2)} give distinct units. (b) Consider T := t 0 + p 2 t and m s (T ) = T 2 -4s and the units of norm s:

E s (T ) = 1 2 T + T 2 -4s ;
they are, for all t, local pth power at p for suitable t 0 depending on p and s, as follows: (i) For t 0 = 0, the units E s (T ) = E s (p 2 t) are local pth powers at p.

(ii) For p ≡ 5 (mod 8), there exist s ∈ {-1, 1} and t 0 ∈ Z ≥1 solution of the congruence t 2 0 ≡ 2s (mod p 2 ) such that the units E s (T ) are local pth powers at p. (iii) We get the data (p = 3, s = -1, t 0 ∈ {4, 5}), (p = 7, s = 1, t 0 ∈ {10, 39}), (p = 11, s = -1, t 0 ∈ {19, 102}), (p = 17, s = -1, t 0 ∈ {24, 265}; s = 1, t 0 ∈ {45, 244}).

As t grows from 1 up to B, for each first occurrence of a square-free integer M ≥ 2 in the factorization m(t) = a 2 p 4 t 2 -2δas = M (t)r(t) 2 (case (a)), or the factorization m(t) = (t 0 + p 2 t) 2 -4s = M (t)r(t) 2 (case (b)), the quadratic fields Q( M (t)), are non p-rational, apart possibly when 1 δ ap 4 t 2 -δs + p 2 t r(t) M (t) ∈ ε p M(t) (case (a)), or

1 2 t 0 + p 2 t + (t 0 + p 2 t) 2 -4s ∈ ε p M(t) (case (b)).
Proof. The case (a) is obvious. Since the case (b) (i) is also obvious, assume t 0 ≡ 0 (mod p 2 ). We have:

E s (T ) 2 ≡ 1 2 T 2 -2s + T T 2 -4s (mod p 2 ), whence E s (T ) 2 ≡ t0 2 √ T 2 -4s (mod p 2 ) under the condition t 2 0 ≡ 2s (mod p 2 ). So, E s (T ) 4 ≡ 1 4 t 2 0 (t 2 0 -4s) ≡ -1 (mod p 2 )
, whence the result. One computes that t 2 0 ≡ 2s (mod p 2 ) has solutions for (p -1)(p + 1) ≡ 0 (mod 16) when s = 1 and (p -1)(p + 5) ≡ 0 (mod 16) when s = -1.

For instance, in case (a) we shall use m (b) has the advantage that the traces of the units are in O(t) instead of O(t 2 ) for case (a).

(t) = p 4 t 2 -s, m(t) = p 4 t 2 -2s, m(t) = p 4 t 2 -4s, m(t) = 9p 4 t 2 -6s, m(t) = 9p 4 t 2 -12s, m(t) = 4p 4 t 2 -2s, m(t) = 25p 4 t 2 -10s, m(t) = 25p 4 t 2 -20s. The case
Since in many computations we are testing if some unit E s (T ) is a global pth power, we state the following result which will be extremely useful in practice because it means that the exceptional cases are present only at the beginning of the F.O.P. list: Theorem 6.4. Let T be of the form T = ct h + c 0 , c ≥ 1, h ≥ 1, c 0 ∈ Z fixed and set T 2 -4s = M (t)r(t) 2 when t runs through Z ≥1 . For B ≫ 0, the maximal bound M pow B of the square-free integers M (t), obtained by the F.O.P. algorithm, for which E s (T ) := For instance T = t 0 + p 2 t, of the case (b) of Theorem 6.3, gives a bound M pow B , of possible exceptional Kummer radicals, of the order of (p 4 B 2 ) 1/p . This implies that when B → ∞, the density of Kummer radicals M such that E s (T ) is not a global pth power is equal to 1. With B = 10 6 , often used in the programs, the bound M pow B tends to 1 quickly as p increases. In practice, for almost all primes p, the F.O.P. lists are without any exception (only the case p = 3 gives larger bounds, as M pow 10 6 ≈ 43267 for the above example; but it remains around 10 6 -43267 = 956733 certified solutions M ). 6.3.2. Program of computation of T K . In case a) of Theorem 6.3, we give the program using together the 16 parametrized radicals and we print short excerpts. The parameter e must be large enough such that p e annihilates T K ; any prime number p > 2 may be illustrated (here we take p = 3, 5, 7). A part of the program is that given in [START_REF] Gras | On p-rationality of number fields[END_REF] for any number field. For convenience, we replace a data of the form [7784110,List([9])], in the outputs, by [7784110, [9]] giving a 3-group T K of Q( √ 7784110) isomorphic to Z/9Z.

6.4. Infiniteness of non p-rational real quadratic fields. All these experiments raise the question of the infiniteness, for any given prime p ≥ 22, of non p-rational real quadratic fields when the non p-rationality is due to R K ≡ 0 (mod p) (i.e., log(ε M ) ≡ 0 (mod p 2 )).

The case p = 2 being trivial because of genus theory for 2-class groups, we suppose p > 2. However, it is easy to prove this fact for p = 2 by means of the regulators.

6.4.1. Explicit families of units. We will built parametrized Kummer radicals and units, in the corresponding fields, which are not pth power of a unit; the method relies on the choice of suitable values of the parameter trace t. This will imply the infiniteness of degree p -1 imaginary cyclic fields of the Section 7 having non trivial p-class group.

Theorem 6.5. (i) Let q ≡ 1 (mod p) be prime, let c / ∈ F ×p q and t q ∈ Z ≥1 such that t q ≡ c 2 + s 2cp 2 (mod q). Then, whatever the bound B, the F.O.P. algorithm applied to the polynomial m(t q + qx) = p 4 (t q + qx) 2 -s, x ∈ Z ≥0 , gives lists of distinct Kummer radicals M , in the ascending order, such that Q( √ M ) is non-p-rational.

(ii) For any given prime p > 2 there exist infinitely many real quadratic fields K such that R K ≡ 0 (mod p), whence infinitely many non p-rational real quadratic fields.

Proof. (i) Criterion of non pth power. Consider m(t) = p 4 t 2 -s and the unit E s (2p 2 t) = p 2 t + p 4 t 2 -s of norm s and local pth power at p. Choose a prime q ≡ 1 (mod p) and let c ∈ Z >1 be non pth power modulo q (whence (q -1) 1 -1 p possibilities). Let t ≡ c 2 + s 2cp 2 (mod q); then: N(E s (2p 2 t) -c) = N(p 2 t -c + p 4 t 2 -s) = (p 2 t -c) 2 -p 4 t 2 + s = c 2 + s -2cp 2 t ≡ 0 (mod q). Such value of t defines the field Q( M (t)), via p 4 t 2 -s = M (t)r(t) 2 , and whatever its residue field at q (F q or F q 2 ), we get E s (2p 2 t) ≡ c (mod q), for some q | qZ; since in the inert case, #F × q 2 = (q -1)(q + 1), with q + 1 ≡ 0 (mod p), c is still non pth power, and E s (2p 2 t) is not a local pth power modulo q, hence not a global pth power.

Corollary 4 . 4 .

 44 Let M ≥ 2 be a given square-free integer and consider the two lists given by the F.O.P. algorithm, for m -1 and m 1 , respectively. Then, assuming B large enough, M appears in the two lists if and only if

  MAIN PROGRAM FOR FUNDAMENTAL UNITS OF NORM s {B=10^7;s=-1;LM=List;for(t=2+s,B,mt=t^2-4*s; M=core(mt);L=List([M]);listput(LM,vector(1,c,L[c]))); VM=vecsort(vector(B-(1+s),c,LM[c]),1,8); print(VM);print("#VM = ",#VM)} s=-1 [M]= [2],[5],[10],[13],[17],[26],[29],[37],[41],[53],[58],[61],[65],[73],[74],[82],[85], [89],[97],[101],[106],[109],[113],[122],[130],[137],[145],[149],[157],[170],[173], [181],[185],[193],[197],[202],[218],[226],[229],[233],[257],[265],[269],[274],[277], [281],[290],[293],[298],[314],[317],[346],[349],[353],[362],[365],[370],[373],[389], (...) [99999860000053],[99999900000029],[99999940000013],[99999980000005]] #VM = 9999742 s=1 [M]= [2],[3],[5],[6],[7],[10],[11],[13],[14],[15],[17],[19],[21],[22],[23],[26],[29],[30], [31],[33],[34],[35],[37],[38],[39],[41],[42],[43],[46],[47],[51],[53],[55],[57],[58], [59],[61],[62],[65],[66],[67],[69],[70],[71],[73],[74],[77],[78],[79],[82],[83],[85], [86],[87],[89],[91],[93],[94],[95],[101],[102],[103],[105],[107],[109],[110],[111], (...) [99999820000077],[99999860000045],[99999900000021],[99999979999997] #VM = 9996610The same program with outputs of the form[M, r, t] for s = 1 gives many examples of squares of fundamental units. For instance, the data[29, 5, 27] defines the unit E 1 (27since 27-2 = 5 2 , then t ′ = 5, r ′ = 1 and E 1

  ,[2],[13],[5],[29],[10],[53],[17],[85],[26],[5],[37],[173],[2],[229],[65],[293],[82], [365],[101],[445],[122],[533],[145],[629],[170],[733],[197],[5],[226],[965],[257], [1093],[290],[1229],[13],

  So, we have to estimate the sums t∈[1,b]

  v_p(#(p-class group))=0 v_p(#(p-torsion group))=1 m(p)=(p+1)^2+1, p=11 M=145 v_p(#(p-class group))=0 v_p(#(p-torsion group))=2 p=16651 M=277289105 v_p(#(p-class group))=0 v_p(#(p-torsion group))=1 m(p)=(p+1)^2-1, p=3 M=15 v_p(#(p-class group))=0 v_p(#(p-torsion group))=1 m(p)=(p+1)^2-4

2 - 1 p

 21 4s may be a pth power in ε M(t) (whence the field Q( M (t)) being prational by exception), is of the order of (c 2 B 2h ) as B → ∞. Proof. Put ε M = 1 2 (a + b √ M ) as usual; then we can write ε M ∼ b √ M and E s (T ) ∼ T so that T and (b √ M ) p are equivalent as M and B tend to infinity; taking the most unfavorable case b = 1, we conclude that M pow B ≪ (c 2 B 2h ) 2/p in general.

7. 2 .

 2 Imaginary cyclic fields with non-trivial p-class group, p > 3. Let χ be the even character of order 2 defining K := Q( √ M ), let p ≥ 3 and let L := K(ζ p ) be the field obtained by adjunction of a primitive pth root of unity; we may assume that K ∩ Q(ζ p ) = Q, otherwise M = p in the case p ≡ 1 (mod 4), case for which there is no known examples of p-primary fundamental unit. Let ω be the p-adic Teichmüller character (so that for all τ ∈ Gal(L/Q), any list of quadratic fields Q( √ M ) obtained by the previous F.O.P. algorithm giving p-primary units E, the ωχ -1 -component of the p-class group of L is non-trivial as soon as E / ∈ ε p M and gives an odd component of the whole p-class group of L. Theorem 7.2. As t grows from 1 up to B, each first occurrence of a square-free integer M ≥ 2 in the factorization m(t) := p 4 t 2 -4s =: M r 2 , the degree p -1 cyclic imaginary subfield of Q( √ M , ζ p ), distinct from Q(ζ p ),has a class number divisible by p, except possibly when the unit E s (p 2 t) := 1 2 [p 2 t + r √ M )] is a p-th power in ε M .

1 .

 1 Introduction and main results 1.1. Definition of the "F.O.P. " algorithm 1.2. Quadratic integers 1.3. Quadratic polynomial units 1.4. Main algorithmic results 2. First examples of application of the F.O.P. algorithm 2.1. Kummer radicals and discriminants given by m s (t) 2.2. Application to minimal class numbers 2.3. Application to minimal orders of p-ramified torsion groups 2.4. Application to minimal orders of logarithmic class groups 3. Units E s (t) vs fundamental units ε M(t) 3.1. Polynomials m s (t) = t 2 -4s and units E s (t) 3.2. Checking of the exponent n in E s (t) = ε n

	M(t)

  Let a ∈ Z ≥1 and δ ∈ {1, 2}. We consider T := 2δ -1 (ap 4 t 2 -δs) and m 1 (T ) giving rise to the unit E 1

	6.3.1. Definitions of local p-th power units. Taking polynomials stemming from suitable
	polynomials m s we can state:
	Theorem 6.3. Let p > 2 be a prime number and let s ∈ {-1, 1}. (a)

The property holds from n = 0 (T(1) =

2), except for M = 5 (T(ε M ) = 1, T(ε 2 M ) = 3) and M = 2 (T(ε M ) = 2, T(ε 2 M ) = 6)).

(ii) Infiniteness. Now, for simplicity to prove the infiniteness, we restrict ourselves to the case m(t) = p 4 t 2 -1 (the case m(t) = p 4 t 2 + 1 may be considered with a similar reasoning in Z[ √ -1] instead of Z). Let ℓ be a prime number arbitrary large and consider the congruence: p 2 (t q + qx) ≡ 1 (mod ℓ); it is equivalent to x = x 0 + yℓ, y ∈ Z ≥0 , where x 0 is a residue modulo ℓ of the constant 1tqp 2 qp 2 ; so, we have p 2 (t q + qx 0 ) -1 = λℓ n , n ≥ 1, ℓ ∤ λ. Computing these m(t)'s, with t = t q + (x 0 + yℓ)q, gives:

the right factor is prime to ℓ; the left one is of the form λℓ n + qyp 2 ℓ, and whatever n, it is possible to choose y such that the ℓ-valuation of λℓ n-1 + qyp 2 is zero. So, for such integers t, we have the factorization m(t) = ℓM ′ r 2 , where M ′ ≥ 1 is square-free and M ′ r 2 prime to ℓ, which defines M := ℓM ′ arbitrary large.

This proves that in the F.O.P. algorithm, when B → ∞, one can find arbitrary large Kummer radicals M (t q + (x 0 + yℓ)q) such that the corresponding unit E 1 (t q + (x 0 + yℓ)q) is a local pth power modulo p, but not a global p-th power.

The main property of the F.O.P. algorithm is that the Kummer radicals obtained are distinct and listed in the ascending order; without the F.O.P. process, all the integers t = t q + (x 0 + yℓ)q giving the same M give E 1 (t q + (x 0 + yℓ)q) = ε n M with n ≡ 0 (mod p). 6.4.2. Unlimited lists of non-p-rational real quadratic fields. Take p = 3, q = 7, c ∈ {2, 3, 4, 5}. With m(t) = 81t 2 -1, then t q ∈ {2, 5}; with m(t) = 81t 2 + 1, then t q ∈ {3, 4} and t = t q + 7x, x ≥ 0. The F.O.P. list is without any exception, giving non 3-rational quadratic fields Q( √ M ) (in the first case, p = 3 is inert and in the second one, p = 3 splits. We give the corresponding list using together the four possibilities: 7. Application to p-class groups of some imaginary cyclic fields Considering, now, the case b) of Theorem 6.3 for p > 2, we use the polynomial m s (T ) = T 2 -4s, with T = t 0 + p 2 t, and the unit of norm s:

for suitable s and t 0 such that E s (T ) be a local pth power at p, which is in particular the case for all p > 2 and all s when t 0 = 0. For t 0 = 0, we get the particular data when the equation t 2 0 ≡ 2s (mod p 2 ) has solutions (which is equivalent to p ≡ 5 (mod 8)):

For p = 2, a "mirror field" may be taken in Q( √ -1, √ M ) (see, e.g., [START_REF] Gras | Sur la norme du groupe des unités d'extensions quadratiques relatives[END_REF] for some results linking 2-class groups and norms of units).

The programs are testing that E s (T ) is not the pth power in ε M .

7.1. Imaginary quadratic fields with non-trivial 3-class group. From the above, we obtain, as consequence, the following selection of illustrations (see Theorem 6.5 claiming that the F.O.P. lists are unbounded as B → ∞): Proof. If E s (t 0 +9t) is not a third power in ε M but a local 3th power at 3, it is 3-primary in the meaning that if ζ 3 is a primitive 3th root of unity, then K(ζ 3 , 3 E s ((t 0 + 9t))/K(ζ 3 ) is unramified (in fact 3 splits in this extension). From reflection theorem (Scholz's Theorem in the present case), 3 divides the class number of Q( √ -3M), even when r > 1 in the factorization m(t) =: M r 2 . The case of t 0 = 0 and s = ±1 is obvious. The second claim comes from Theorem 6.5 (see numerical part below).

Program for lists of 3-class groups of imaginary quadratic fields.

Note that the case where E s (t 0 + 9t) is a third power is very rare because it happens only for very large t 0 + 9t giving a small Kummer radical M . One may verify the claim by means of the following program, in the case s = -1 valid for all t 0 , where [M, Vh] gives in Vh the 3-structure of the class group of Q( √ -3M ); at the end of each output, one sees the list of exceptions (case of third powers), where the output [M, n] means that for the Kummer radical M = M (t), then E -1 (t 0 + 9t) = ε n M . We may see that any excerpt for t large enough give no exceptions:

LISTS OF 3-CLASS GROUPS OF IMAGINARY QUADRATIC FIELDS {p=3;B=10^5;L3=List;Lh=List;Lt0=List ([0,4,5]);for(t=1,B, for(ell=1, 3,t0=Lt0[ell];mt=(t0+9*t)^2+4;ut=(t0+9*t)/2;vt=1/2; C=core(mt,1);M=C [1];r=C [2];res=Mod (M,4) B ≈ 9321.76 give a good verification of the Heuristic 6.4. This also means that all the integers M larger than 9029 leads to non-trivial 3-class groups, and they are very numerous !

We note that some M 's (as 29, 74, 82, 85, . . .) are in the list of exceptions despite a non-trivial 3-class group; this is equivalent to the fact that, even if E -1 (t 0 + 9t) ∈ ε 3 M , either the 3-regulator R K of K is non-trivial or its 3-class group is non-trivial.

In this interval, all the 5-class groups obtained are non-trivial, except for s = -1 and M = 29, then for s = 1 and M = 21. From Remark 1.5, we compute: log(155625629)/ log (5 4 • 25 • 10 4 ) ≈ 0.99978777. Theorem 6.4 gives possible exceptions up to M Consider the case p = 7, s ∈ {-1, 1}; exceptionally, we give the complete lists:


Fabrizio Ameli
email: fabrizio.ameli@roma1.infn.it

Stefano Russo
email: stefano.russo@na.infn.it

Gabriele Giovanetti
email: ggiovanetti@gmail.com

Fabrice Gensolen
email: gensolen@cppm.in2p3.fr

Performance results of a prototype board for copper data transmission

The experience gained in designing submarine neutrino telescopes suggested to explore new ways of realizing the data transmission backbone at the detection unit level. In order to decrease the difficulties of integration and handling of the backbones, some effort has been spent in developing a backbone based on copper links with simple tracts of cable connecting contiguous storeys. This work is aimed at the presentation of the general architecture of the system, at the description of an electronic board prototype designed to test the project feasibility with the first results obtained. The main goal of the experimental setup was measuring the recovered clock jitter under various conditions, with and without cables. The jitter measured on the cleaned clock amounts to hundreds of picoseconds, well below the sub-nanosecond time resolution required by this kind of experiments.

I. INTRODUCTION

Lessons learnt from current experiments developing neutrino telescopes, led us to look for new ways of implementing data transmission at the detection unit (DU) level. The use of fiber optics in backbones usually implies high cost mainly due to constraints of maintaining the optical power budget. This implies a high cost of cables and connectors, substantial manpower effort for the integration (fiber handling is a delicate operation), the difficulty of testing the integrated system, the power required by electro-optical transceivers and, in the case of DWDM systems, the high price of the transceiver itself. Most of these issues are circumvented through the use of copper links: handling -and hence integration -is much easier, connectors are more common and cheaper than the optical counterpart, while the devices which implement the physical layer of the transmission system are inexpensive. On the other hand, the use of copper links requires a sophisticated implementation of the transmission system, both from a hardware and software point of view.

Recent technological progress, specifically in audio/video data transmission, made available on the market suitable devices capable of transmitting high data rates (many Gb/s) over long distances (hundreds of meters, depending on the medium). The idea behind this paper is to adapt such a proven technology for the purposes of a submarine experiment.

The basic unit of neutrino telescopes can be defined as the DU. The DU consists of many "storeys" vertically distributed in a structure as high as 800 m; each storey hosts photomultipliers (PMTs), usually grouped in 3 to 6 elements. The total number of sensors per DU ranges from 60 to 120. The telescope is far from the shore laboratory, typically it is placed at a depth of thousands of meters and at distances of many tens of kilometers, and produces an amount of data on the order of 1 Gb/s; these constraints require a telescope-to-shore data transmission system based on fiber optics. The data produced by a group of sensors are collected by the storey electronics and routed to on-shore through a DU backbone. The transmission backbone inside the DU has been successfully implemented in optical fibers in NEMO and Antares experiments [START_REF]NEMO website[END_REF], [START_REF]ANTARES website[END_REF], [START_REF] Ameli | The data acquisition and transport design for nemo phase 1[END_REF], where DWDM backbones based on Add and Drop modules have been realized.

In Section II the requirements of a copper system will be briefly discussed and, in Section III, both a possible implementation of the general architecture and the prototype board designed to verify the feasibility of the system are presented. After a description of the experimental setup in Section IV, preliminary results measured on the prototype board will be shown. Finally in Section V some conclusions are drawn on the performance of the system and on its applicability.

II. COPPER LINK HIGHLIGHTS

The short distances (maximum 50 m) between storeys and the relatively low data rate per storey led us to evaluate a new approach based on copper wires in the DU backbone. The main constraints taken into consideration in the design of a copper backbone were:

• a maximum inter-storey distance of 50 m;

• a reasonable data rate on the link is about 1 Gb/s; • that the cable should be mechanically manageable;

• that storeys be connected by single tracts of cables. The data rate over the backbone is highly asymmetrical: data flowing from on-shore to the storeys are for control and setup, i.e. the data rate can be as low as few hundreds of kb/s per storey. On the opposite direction the data rate must support the transmission of physics data, at least 10 Mb/s per 10" PMT. For both directions we assume that the best (and only choice, if minimization of cable number is required) is to transmit a synchronous bitstream which embeds both clock and data. The low speed of the control channel allows the receiver to recover the clock signal guaranteeing the necessary sub-nanosecond precision required by this kind of experiment.

A low wire count is necessary in order to have a manageable submarine cable. The requirment that storeys should be connected with independent tracts, without signal extraction from the backbone by means of breakouts, aids in this respect: connectors can be realized with a low number of electrical pins.

III. SYSTEM ARCHITECTURE

The general architecture of the copper backbone has been already explained in other documents [START_REF] Ameli | Km3net: A proposal design for a detection unit data transmission system based on a copper backbone[END_REF]. Briefly, the daisy chain architecture, shown in fig. 1, requires that each node receives the slow serial data stream transmitted by the previous node on the blue link, recovers the clock embedded in the stream, extracts clock and data, cleans the clock and uses this de-jittered clock to transmit data to the next node in the chain and to provide the clock reference for transmission in the other direction. Of course, the clean clock is also used to feed each electronic device which needs a reference synchronous to the backbone clock. The Copper Node hosts an FPGA so that the daisy chain mechanism is completely user transparent: each node in the chain has to manage all the data flow and can "add and drop" its own payload; the node appears as a parallel interface to transmit and receive data. There are also controls and clock flags to manage timing issues and handshake protocols. This document describes the preliminary tests of the reduced Copper Node prototype board, the so called "cNodeV1" board. Actually the cNodeV1 board is a subset of a complete copper node which would be able to provide the required full duplex, asymmetrical full daisy chained communication link on copper media. The cNodeV1 implements only the slow path of the chain: data are serially transmitted at a rate of about 200 Mb/s.

The main goal of the prototype is to demonstrate that the full daisy chain scheme is capable to transmit serially on a copper medium both clock and data, that the Bit Error Rate (BER) on data is acceptable, the clock is recoverable, and its quality is still adequate for neutrino experiments. This last requirement imposes a maximum jitter on the clock well below 1 ns. This being the major issue regarding the data transmission on copper, the cNodeV1 board is intended to give an answer to this prolem.

To avoid the constraint to choose in advance a specific transmission cable, i.e. the characteristic cable impedance, the line interface has not been embedded on the board. Instead, use of high speed SMA connectors allows to change both the line driver and equalizer and also the cable. Moreover we do not address here the problems concerning either the impedance change, due to the high pressure in the submarine environment typical of a neutrino telescope, or the behavior of the system with real submarine connectors. Figure 2 shows a block diagram of the cNodeV1 board with the line interface.

IV. EXPERIMENTAL SETUP

The cNodeV1 can be plugged into a Xilinx ML50x evaluation board using an adapter already developed for another application: this configuration is shown in fig. 3. For the tests we used both a ML507 board, which hosts a Virtex-5 FX FPGA with a hardwired embedded PowerPC processor, and a ML505 board, where a MicroBlaze embedded processor has been synthesized in the Virtex-5 LX FPGA. To measure the BER, a dedicated module has been specifically written: the FPGA transmits a sequence of random bytes, coded according to 8b10b protocol, to the transmitter which serializes them; after the cable, the stream is received by the receiver, de-serialized and passed back to the FPGA. After an initial phase, the FPGA logically locks onto the received stream and transmitted data are compared against received data: in case of errors a counter is incremented and updated each second. A program has been written to manage the system: the cNodeV1 board can be intialized, the BER module can be switched on and off or reset, errors on the transmission can be forced to check if the BER measure is effective 1 , and the BER results can be read by the processor. A simple RS232 connection is used for the communication.

All the measures have been taken using a LeCroy SDA Zi760, with 6 GHz of bandwidth and 40 GS/s of sampling rate. This instrument is specifically designed for serial data analysis allowing the user to extract bitstream feature and characterizing the clock performance of the system independently from the receiver.

A. Single hop setup

The single hop configuration consists of one cNodeV1 board plugged into an ML505 board. Fig. 4 schematically shows the block diagram of the system. The clock used for transmission is a local 16.384 MHz oscillator. The receiver, using an internal Clock and Data Recovery (CDR) module, extracts from the stream both clock and data, which are acquired by the FPGA. The recovered clock is also fed to the PLL which provides a synchronous and clean version of it.

In the next Sections the results obtained under different line conditions will be analyzed.

1) Single hop with coaxial cables pair:

In this configuration data are transmitted to the receiver through high speed coaxial cables pair 20 cm long without inserting any driver or equalizer: this is the best possible condition which gives us an idea of the minimum jitter value attainable. The waveforms of the source clock and of the clocks before (CDR Clock) 1 This is particularly useful when the link is too good to show errors in a reasonable time The standard deviation of the TIE for the source clock is about 340 ps; even in this optimal case the TIE of the recovered clock before been cleaned has a standard deviation of 450 ps while, after applying the PLL, the jitter is reduced down to 10 ps.

2) Single hop with 2 m long CAT5 cable: With this setup, a CAT5 cable 2 m long has been inserted between the line driver and the receiver; the TIE standard deviation of the clock cleaned by the PLL is slightly worse than what we found previously in Section IV-A1 but it is still very good being about 13 ps for the clean clock. The TIE of the recovered clock is now 500 ps due to the longer cable.

3) Single hop with 30 m long CAT5e cable: The same performance values have been evaluated using a CAT5e cable 30 meters long. The TIE standard deviation for the clean clock is still about 15 ps, while the value for the recovered clock increased up to 550 ps.

B. Double Hop setup

In the double hop configuration depicted in fig. 5 we have one cNodeV1 prototype board plugged into the ML505 board and a second cNodeV1 which, plugged into the adapter, can work in a stand-alone fashion. Data are transmitted by the first node to the second which recovers the clock, cleans it by means of the PLL and retransmits data back to the first node which, in turn, recovers and cleans the clock. Measures are made to assess system synchronicity and clock jitter.

1) Double hop with 300 m long CAT5e cable and short coaxial cables: In the examined configuration, the first hop uses a 300 m long CAT5e cable with a driver and equalizer pair; the second hop is covered by 2 high speed coaxial cables 1 m long without any driver or equalizer. Each node recovers the line clock and cleans it by means of its own PLL: the clock from the first node shows a standard deviation of the TIE of about 19 ps while the clock from the second node has a value of 37 ps. The mean period for the two clocks is the same: 61.03563 ns (calculated over about 8 million time windows 50 µs each) with a standard deviation of 6 ps and a maximum peak-to-peak deviation of 90 ps. This confirms that the two nodes are synchronous and the instantaneous deviation is small.

C. Jitter analysis performed by SDA analyzer

We tried also to estimate the BER on data letting the SDA analyzer do the job for us. We transmitted the random data on the 300 m long CAT5e cable using both driver and equalizer; then the two differential signals driven by the equalizer were read by the instrument and subtracted to yield the single ended serialized stream. Fig. 6 shows the jitter analysis performed on this signal: the eye diagram at a BER value of 10 -12 is still open by a 20 % of the unit interval, as extrapolated in the bathtub curve. The TIE time evolution does not exhibit any evident trend and the TIE histogram looks gaussian, stressing that there are no deterministic jitter components. The estimated total jitter on the signal is less than 4 ns. The clock extracted from the stream will be cleaned using the PLL to provide the performance stated in IV-B1.

V. CONCLUSION

The preliminary results shown in this paper are very encouraging: the transmission of a 200 Mb/s stream on a CAT5e cable seems feasible even over distances longer than 300 m. Data received show a low BER value; the clock can be extracted from the serial stream and, after a cleaning step implemented by a PLL, the jitter is reduced to tens of ps. The tests used only two hops, but in the near future a higher number of boards will be cascaded to check the performance of a longer chain.

The effects of change in cable impedance due to the pressure of submarine environment coupled with the use of inexpensive connectors -not normally deemed suitable for use with fast differential signals -will be analyzed in the near future using a hyperbaric chamber and a setup based on commercial cables and connectors. Since the maximum distance between adjacent storeys is expected to be 50 m, we are optimistic that the performance at high pressures will be similar to that measured in this work, using standard connectors and cables within the 300 m length already studied.

Figure 1 .

 1 Figure 1. Full Double Daisy Chain architecture.

Figure 2 .

 2 Figure 2. Block diagram of the Master Chain block shown in fig. 1. Also the line interface is present with a cable driver and an equalizer.

Figure 3 .

 3 Figure 3. Experimental setup with the cNodeV1 board plugged into the ML505 board through the adapter. The line driver transmits the signal through a CAT5e cable 30 meters long and the signal is received by the cNodeV1 receiver after the equalization.

Figure 4 .

 4 Figure 4. Single hop block diagram: the cNodeV1 board, plugged into the host board, drives the cable and receives back the data.

Figure 5 .

 5 Figure 5. Double hop block diagram: the first hop consists of a cNodeV1 plugged into the ML505 board and acting as the chain source. The second hop is implemented with a cNodeV1 board working stand-alone. The driver and equalizer pair are inserted only in the path between first and second node.

Figure 6 .

 6 Figure 6. Jitter analysis provided by the LeCroy SDA instrument: the line differential signals after the driver and equalizer pair and a CAT5e cable 300 m long are analyzed.

The TIE is a first rough measure of the jitter: it represents the distance between the edges of the acquired clock and the expected edges inferred from the clock itself. See for example[START_REF]Understanding and characterizing timing jitter[END_REF]

ACKNOWLEDGMENTS

This work is supported through the EU-funded FP6 KM3NeT Design Study Contract No. 011937.


New Social Contract for Ukraine https://voxukraine.org/en/new-social-contract-for-ukraine Dr Violeta MOSKALU, founder of Global Ukraine Foundation Moskalu, V. (2015), New Social Contract for Ukraine, VoxUkraine, https://voxukraine.org/en/new-social-contract-for-ukraine/.

May 2015

A systemic analysis of the Ukrainian political crisis shows that inclusive leadership is the key success factor of reforms in Ukraine. Post-Maidan Ukraine needs a New Deal (a kind of agreement between civil society, business and government, a new social pact to build a modern state), centered around the rule of law and the effective management tools.

In Western democracies, the three main categories of actors (power, business, and civil society) are separated, and there are clear interaction rules to assure their cooperation and to provide a framework in order to avoid conflicts of interest. From this point of view, the real problem in reforming the Ukrainian political system is the need to cut the Gordian knot of political-oligarchic system where financial resources, political parties and the media are in the hands of the same people or clans.

Ukrainian crisis -how to change the system?

The oligarchic political system is a source of tension and the origin of a strong sense of injustice in society. That is why it is urgent and critical to dismantle this oligarchic system in order to save the country. New rules must elevate the Ukrainian political life to Western standards based on fundamental democratic principles that aim to reduce the conflicts of interests within the society. For this, the three branches of governmentlegislative, executive and judicial -should also be completely separated. Most of the necessary changes are known by the Ukrainians, the main points were even fixed in the coalition agreement signed after the parliamentary elections in October 2014.

The issue of injustice and the inequality in the distribution of national wealth is one of the key issues for changing the rules and reforming the old corrupted system that is why fighting massive corruption at the highest level is priority number one. The famous economist and Nobel laureate Joseph Stiglitz noted in his latest book that the issue of inequality and injustice is a political choice, the result of the accumulation of unfair political decisions and misplaced priorities.

How then to explain the difficulties of the realization and implementation of the coalition agreement? Why are reforms not implemented even though everyone knows WHAT should be done? The answer is simple -the resistance to change expressed by the old system. The resistance to change is a known phenomenon; there is even a specific scientific approach and particular understanding of the key success factors by the times of change. For successful change management in this reforms period, the «resistance phenomenon» should be taken into account. Certain people fear changes because there are some instinctive psychological reactions. Fear of losing certain benefits (resources, influence, power, social status, etc.) explains the strong resistance to change.

System analysis of post-Maidan Ukrainian political situation shows clearly a separate category of «conservative» actors (certain representatives of the old political parties of the past, some bureaucrats and officials who are part of the system that needs to be changed), who talk a lot about reforms, but do little or nothing (particularly in the fight against corruption or purification of the judiciary system). The paradox is that on the one hand, they can be able to take some initiatives for changes (or at least to sham it), but as long as they try to «win something» or «to avoid any losses», the final results will be very disappointing, because too much energy is directed to support their own ambitions or to strengthen personal positions. The last Ukraine -EU summit stressed the absolute necessity to begin finally doing real reforms, because otherwise Ukraine risks losing the support of Western partners in the near future.

Meanwhile, most of the civil society is more mature than the political elite. Civil society has learned to observe and analyze real facts and concrete actions, rather than to rely on political promises. Thanks to new information and communication technologies, despite the fact that traditional media are under the influence of oligarchic clans, conscious citizens instantly realize everything that is done or not done by political leaders. The government did not know how (or did not want?) to restore public confidence in state institutions, the economic situation has deteriorated significantly, and the war in the East can no longer serve as an universal excuse. A careful analysis and comparison of obligations (in words) and real acts of post-Maidan political leaders shows the weakness of our political elites. Or their weak will to respect and to fulfill their recent political promises. Political will -which is defined as the «will to act», the «ability to act» and the «feeling that we are required to act» -is critical for real changes to occur.

The situation may seem paradoxical: the elite do not want real changes, to avoid losses of their benefits; the business community and civil society understand that changes are inevitable. As publicly stated by former Finance Minister Oleksandr Shlapak (2014), the «political will exists within civil society», but the political will is not strong enoughamong the political elite, as we have seen through the lack of real actions, such as fight against corruption (on high political level). close ties and strong relationships among political elites make implementing and enforcing policy changes particularly challenging.

The two key factors for successful changes are synergy research and inclusive leadership.

The Post-Maidan volunteer phenomenon shows that in Ukraine more and more people are mobilized to solve problems at local and national level, or in the promotion of the interests of Ukraine in the world. They want to act and they feel the need or even the duty to act, and they finally will change the system. The Ukrainian volunteers, or the phenomenon of «the-do-it-yourself-country» are the proof that in Ukraine there are more and more responsible citizens. It only remains to spread the momentum of solidarity and collective action. The basic rule of change management, the core idea of the famous French author Jean-Christian Fauvet is very simple: those who really want changes should look for allies, because very often we lose not because of the opponents, but due to lack of allies.

That is why constant communication is needed in order to involve all potential allies in the project, to develop trust and collective action synergy based on cooperation. A very important thing: trust does not mean "don't attack", but rather "don't defend yourself". As long as the other person has done nothing to lose trust, he/she can be treated as trustful.

At the same time, any government will resist creating tools to limit its power. Civil society, the intellectual and business environment should show an example of coordination, as during the Maidan. They must create the conditions of an "intellectual and economic Maidan".

Inclusive leadership and human capital investments for successful reforms in

Ukraine

Post-Maidan Ukraine needs a New Deal (a kind of agreement between civil society, business and government, a new social pact to build a modern state), centered around the rule of law and the effective management tools. The key words for this New Deal System are cooperation and trust. Because knowledge economy in the 21st century is built mainly through intellectual capital, trust and credibility are the central elements that are required for the development of collective intelligence. A series of scientific papers focused on the issue of trust and its determinants, the factors that «create» trust. The researchers highlighted the qualities that people must have in order to generate trust [START_REF] Coleman | Social Capital in the Creation of Human Capital[END_REF][START_REF] Mcallister | Affect -and cognition -based trust as foundations for interpersonal cooperation in organizations[END_REF]. These factors are competence, transparency, honesty and support (or compassion).

  Intellectual capital consists of human capital (knowledge and skills that belong to each individual) and social capital (based on trust that enables people to cooperate). Strong social capital makes it easier for people to share their ideas and knowledge and finally creates needed conditions for the development of collective intelligence. The centerpiece of this new paradigm -thinking about collective intelligence and invest in human capital to succeed in the globalization era where cooperation and trust are important components of economic value-added (cf. diagram). © Moskalu, Violeta (2012), A socio-cognitive analysis of the employee ownership impact on value creation of French Public Companies, PhD in Management Sciences, University of Lorraine, Metz-Nancy, France. Therefore, in the actual context of changes within the current political and socio-economic system in Ukraine, successful leaders are the ones who quickly realize that times have changed and there is no other choice but to match the expectations of society. The lack of competence, transparency and integrity will significantly reduce the duration of political careers, because 21st century's basic capital will be the leader's authenticity. «Say what you do and do what you say,» -if you want that people believe you, this will be the new political life postulate. Times change, and Ukraine is changing. Now there is a great demand for the new elite to take initiative and assume political and administrative responsibility for the future development of Ukraine. Ukraine needs competent, transparent, honest, sincere, and especially caring leaders who are able to put the collective interest and public service above their personal interests. As the famous English statesman Benjamin Disraeli wrote, «a politician thinks of the next election and a statesman thinks about future generation».