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Roland Queme

Publications and source records attributed to Roland Queme.

15 recordsLinked to original sources
arXiv2011

Complements on Furtwängler's second theorem and Vandiver' s cyclotomic integers

This article deals with a conjecture generalizing the second case of Fermat's Last Theorem, called $SFLT2$ conjecture: {\it Let $p>3$ be a prime, $K:=\Q(ζ)$ the $p$th cyclotomic field and $\Z_K$ its ring of integers. The diophantine equation $(u+vζ)\Z_K=\mk w_1^p$, with $u,v\in\Z\backslash\{0\}$ coprime, $uv\equiv 0 \bmod p$ and $\mk w_1$ ideal of $\Z_K$, has no solution.} Assuming that $SFLT2$ fails for $(p,u,v)$, let $q$ be an odd prime not dividing $uv$, $n$ the order of $\frac{v}{u}\bmod q$, $ξ$ a primitive $n$th root of unity and $M:=\Q(ξ,ζ)$. The aim of this complement of the article [GQ] of G. Gras and R. Quême on the same topic, is to exhibit some strong properties of the decomposition of the primes $\mk Q$ of $\Z_M$ over $q$ in certain Kummer $p$-extensions of the field $M$, to derive from them a weak conjecture which implies that the SFLT2 equation can always take the reduced form $u+ζv\in K^{\times p}$ and to set a conjecture implying SFLT2.

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arXiv2009

Singular integers and p-class group of cyclotomic fields

Let $p$ be an irregular prime. Let $K=\Q(ζ)$ 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.

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arXiv2007

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 $K=\Q(ζ_p)$ 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 $h^-(K)$.

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arXiv2007

Singular Integers and Kummer-Stickelberger relation

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 a p-cyclotomic field and O_K be its ring of integers. Let pi be the prime ideal of K lying over p. Let sigma : zeta --> zeta^v be the Q-isomorphism of K for a primitive root v mod p. The subgroup of exponent p of the class group of K can be seen as a direct sum oplus_{i=1}^r Gamma_i of groups of order p where the group Gamma_i is annihilated by a polynomial sigma-mu_i with mu_i \in F_p^*. Let Gamma be one of the Gamma_i. From Kummer, there exist prime non-principal prime ideals Q of inertial degree 1 with Cl(Q) \in Gamma. We show that there exists singular semi-primary integers A such that A O_K= Q^p with A^{sigma-mu} = alpha^p where alpha in K. Let E = \frac{alpha^p}{\barα^p}-1 and nu the positive integer defined by nu = v_pi(E) -(p-1), where v_pi(.) is the pi-adic valuation. The aims of this article are to describe the pi-adic expansions of the Gauss Sum g(Q) and of E and to derive an upper bound of nu from the Jacobi resolvents and Kummer-Stickelberger relation of the cyclotomic field K.

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arXiv2007

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.

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arXiv2006

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.

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arXiv2006

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(σ)} if q = 1 (mod p), 2) pi^{2p-1} || A^{P(σ)} if q = 1 (mod p) and p^{(q-1)/p} = 1 (mod q). 3) pi^{2p} | A^{P(σ)} 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.

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arXiv2006

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.

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arXiv2005

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.

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arXiv2005

pi-adic approach of p-class group and unit group of p-cyclotomic fields

Let p > 2 be a prime. Let Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of Q(zeta) lying over p. This article aims to describe some pi-adic congruences characterizing the structure of the p-class group and of the unit group of the field Q(zeta). For the unit group, this article supplements the 1954 and 1956 papers of Denes on this topic. A complete summarizing of the results obtained follows in the Introduction section of the paper (pages 3 to 6). This new version of the article with the same title, submitted with reference math.NT/0407430 25 Jul 2004: - corrects several typing errors in the introduction and in the paper, - simplifies some proofs of pi-adic congruences connected to p-class group, - removes the section dealing of the singular group foreseen in an independant paper.

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arXiv2003

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

This paper is a contribution to the description of some congruences on the odd prime factors of the class number of the number fields. An example of results obtained is: Let L/Q be a finite Galois solvable extension with [L:Q]=N, where N > 1 is odd. Let h(L) be the class number of L. Suppose that h(L) > 1. Let p be a prime dividing h(L). Let r be the rank of the p-class group of L. Then the product p\times (p^r-1)\times(p^{r-1}-1)\times ....\times (p-1) and N are not coprime. The proofs are elementary.

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arXiv2002

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.

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arXiv2002

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 $\Q(zeta_p)$.

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arXiv1999

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.

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