Some congruences on prime factors of class number of algebraic extensions K/Q

This paper is an updated version of ANT-0372 (2002 dec 4) with the same title. Several errors are corrected in this version. An example of the kind of results obtained is: Let K/\Q be an abelian extension with N = [K:\Q] > 1, N odd. Let h(K) be the class number of K. Suppose that h(K) > 1. Let p be a prime dividing h(K). Let r_p be the rank of the p-class group of K. Then p \times (p^{r_p}-1) and N are not coprime. The paper is at elementary level and contains a lot of numerical examples

On pi-adic expansion of singular integers of the p-cyclotomic field

Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K = Q(zeta) be the p-cyclotomic field and let O_K be the ring of integers of K. Let pi be the prime ideal of K lying over p. An integer B \in O_K is said singular if B^{1/p} not \in K and if B O_K = b^p where b is an ideal of O_K. An integer B \in O_K is said semi-primary if B = beta mod pi^2 where the natural beta is coprime with p. Let sigma be a Q-isomorphism of the field K generating the Galois group Gal(K/Q). When p is irregular, there exists at least one subgroup Gamma of order p of the class group of K annihilated by a polynomial sigma - mu with mu \in F_p^*. We prove the existence, for each Gamma, of singular semi-primary integers B where B O_K= b^p with class Cl(b) \in Gamma and B^{sigma-mu} \in K^p and we describe their pi-adic expansion. This paper is at a strictly elementary level.Comment: A correction in proof of lemma 2.2 p. 3 is mad

Some questions on the class group of cyclotomic fields

This article deals with a study of the structure of the class group of the cyclotomic field K=\Q(\zeta_p) for $p$ an odd prime number, starting from Stickelberger relation. The present state of this work leads me to set a question for all the prime numbers $h\not=p$ which divide the relative class number $h^-(K)$.Comment: The title is modified. The section 4 is removed because it contains an irrecoverable error: It is not possible to use epsilon_chi U_p instead of epsilon_chi U. The section 4 is consequently remove

On the class number of cyclic extensions K/Q

Let K/Q be a cyclic extension. In this paper, we give several congruences connecting the prime divisors of the degree g= [K:Q] with the prime divisors of the class number h of K/Q. As an exemple, the theorem: Let K/Q be a cyclic extension with [K:Q]=g. Suppose that g is not divisible by 2 . Let g_j, j=1,...m, be the prime divisors of g. Let h_i, i=1,...r, be the prime divisors of the class number h of K/Q. If for one prime factor h_i of h, the h_i-component G(h_i) of the class group G of K/Q is cyclic then: else h_i divides g, else h_i = 1 (mod g_j) for at least one prime divisor g_j of g. The results obtained are all in accordance with class number tables of Washington, Masley, Girtsmair, Schoof, Jeannin and number fields server megrez.math.u-bordeaux.fr The proofs are strictly elementary

Singular numbers and Stickelberger relation

Let p be an odd prime. Let K_p = Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of K_p lying over p. Let G be the Galois group of K_p. Let v be a primitive root mod p. Let sigma be a Q-isomorphism of K_p. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 in Z[G], where v^n is understood (mod p). We apply Stickelberger relation to odd prime numbers q different of p and to some singular integers A of K_p connected with the p-class group C_p of K_p and prove the pi-adic congruences: 1) pi^{2p-1} | A^{P(\sigma)} if q = 1 (mod p), 2) pi^{2p-1} || A^{P(\sigma)} if q = 1 (mod p) and p^{(q-1)/p} = 1 (mod q). 3) pi^{2p} | A^{P(\sigma)} if q not = 1 (mod p). These results allow us to connect the structure of the p-class group C_p with pi-adic expression of singular numbers A and with solutions of some explicit congruences mod p in Z[X]. The last secion applies Stickelberger relation to describe the structure of the complete class group of K_p.Comment: Some congruences for class number of quadratic fields and biquadratic fields contained in cyclotomic fields are derived of Stickelberger relation in section 7 p. 2

Singular integers and p-class group of cyclotomic fields

Let $p$ be an irregular prime. Let K=\Q(\zeta) be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions S/\Q of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$. We give an algebraic construction of the subfields $M$ of $S$ with degree [M:\Q]=p and an explicit formula for the prime decomposition and ramification of the prime number $p$ in the extensions $S/K$, M/\Q and $S/M$. In the last section, we examine the consequences of these results for the Vandiver's conjecture. This article is at elementary level on Classical Algebraic Number Theory.Comment: The section 7 on the consequences of the previous sections of the article on the Vandiver's conjecture contains an error and is remove

On prime factors of class number of cyclotomic fields

Let p be an odd prime. Let K = \Q(zeta) be the p-cyclotomic number field. Let v be a primitive root mod p and sigma : zeta --> zeta^v be a \Q-isomorphism of the extension K/\Q generating the Galois group G of K/\Q. For n in Z, the notation v^n is understood by v^n mod p with 1 \leq v^n \leq p-1. Let P(X) = \sum_{i=0}^{p-2} v^{-i}X^i \in \Z[X] be the Stickelberger polynomial. P(sigma) annihilates the class group C of K. There exists a polynomial Q(X) \in \Z[X] such that P(sigma)(sigma-v) = p\times Q(sigma) and such that Q(sigma) annihilates the p-class group C_p of K (the subgroup of exponent p of C). In the other hand sigma^{(p-1)/2}+1 annihilates the relative class group of K. The simultaneous application of these results brings some informations on the structure of the class group C, give some explicit congruences in \Z[v] mod p for the p-class group C_p of K and some explicit congruences in \Z[v] mod h for the h-class group of K for all the prime divisors h \not = p of the class number h(K). We detail at the end the case of class number of quadratic and biquadratic fields contained in the cyclotomic field K and give a general MAPLE algorithm.Comment: The numerical MAPLE algorithm of the section 5 p. 18 is adde

Some applications of Kummer and Stickelberger relations

Let p be an odd prime. Let K = Q(zeta) be the p-cyclotomic field. Let v be any primitive root mod p. Let sigma be a Q-isomorphism of K. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 \in Z[G] where 1 \leq v^n \leq p-1 is a notation mod p. We apply a Kummer and Stickelberger relation of K to some singular not primary numbers A of K connected to p-class group C_p of K and prove they verify the congruence A^P(sigma) = 1 mod p^2. This p-adic method on singular numbers A allows us to prove: in a straightforward way the connection between relative p-class group C_p^- and the solutions of some explicit congruences mod p in Z[X]: \sum_{i=1}^{p-2} ((v^{-(i-1)} - v^{-i} v) /p) X^{i-1} \equiv 0 mod p and that if (p-1)/2 is odd then the Bernoulli Number B_((p+1)/2) not = 0 mod p. In this version some congruences deduced of Stickelberger relation for prime ideals Q of K of inertial degree f > 1 are added.Comment: The sufficient condition for equality of the rank of the relative p-class group C_p^- and the index of irregularity i_p of K added in previous version not correct is removed of this version. In this version some congruences deduced of Stickelberger relation for prime ideals Q of K of inertial degree f > 1 are adde

On Kummer and Stickelberger relations

Let p be an odd prime. Let K_p = \Q(zeta_p) be the p-cyclotomic field. We apply a Kummer and Stickelberger relation of K_p to some singular not primary numbers A of K_p connected to p-class group of K_p and prove they verify the congruence A = 1 mod p^2. Let v be a primitive root mod p. This p-adic improvement on singular numbers A allows us to connect in a straightforward way the p-class group C_p to the solutions of some explicit congruence mod p: \sum_{i=1}^{p-2} X^{i-1} \times (\frac{v^{-(i-1)}-v^{-i}\times v}{p}) \equiv 0 mod p: where X is a natural integer and where v^n is understood as v^n mod p with 1 \leq v^n \leq p-1 with n integer \in \Z. The numerical verification of this congruence is completely consistent with table of irregular primes in Washington p. 410

A classical approach on cyclotomic fields and Fermat-Wiles theorem

This paper is submitted to Algebraic-Number-Theory Archives for validation by Number Theorists Community. It is an update of the previous versions ANT-0155, ANT-0170, ANT-0205, ANT-0237, ANT-0321, ANT-0333, and ANT-0356, of which the first four were titled `A generalization of Eichler criterium for Fermat's Last Theorem' and the last three were titled `A classical approach on Fermat-Wiles theorem'. This version contains a complete reorganization of the paper with a first part dealing with cyclotomic fields (independently of FLT) from page 10 to 72 and a second part dealing with FLT from page 73 to end. This version improves our previous results on cyclotomic fields and contains several significant error corrections. The proofs rest on classical theory of cyclotomic number fields \Q(zeta_p)