2013/04/23 by Roland Quême, Quême, Roland
Mathematics · #History and Theory of Mathematics #Mathematics and Applications #math.NT #msc:11D41 #msc:11R18 #msc:11R37
paper · pdf · doi:10.48550/arxiv.1304.6179
13 pages; this article is a part of the restructuration of the article : Complements on Furtwängler's second theorem and Vandiver's cyclotomic units, arXiv 1109.0956 (2011)
arxiv created 2013/04/23 · arxiv updated 2013/04/24
This article, complement to the article [Que], deals with some generalizations of Futwängler's theorems for the second case of Fermat's Last Theorem (FLT2). Let p be an odd prime, ζ a pth primitive root of unity, K:=\Q(ζ) and CℓK the class group of K. A prime q is said p-principal if the class cℓK (\mk qK)∈ CℓK of any prime ideal \mk qK of \ZK over q is the pth power of a class. Assume that FLT2 fails for (p,x,y,z) where x, y, z are mutually coprime integers, p divides y and xp+yp+zp=0. Let q be a prime dividing ((xp+yp)(yp+zp)(zp+xp))/((x+y)(y+z)(z+x)) and \mk qK be any prime ideal of K over q. We obtain the p-power residue symbols relations: ((p)/(\mk qK))K=((1-ζj)/(\mk qK))K for j=1,…,p-1. As an application, we prove that: if Vandiver's conjecture holds for p then q is a p-principal prime. Similarly, let q be a prime dividing ((xp-yp)(yp-zp)(zp-xp))/((x-y)(y-z)(z-x)) and \mk qK be the prime ideal of K over q dividing (xζ-y)(zζ-y)(xζ-z). We give an explicit formula for the p-power residue symbols (\fracεk\mk qK)K for all k with 1<k≤(p-1)/(2), where εk is the cyclotomic unit given by εk=:ζ(1-k)/2⋅(1+ζk)/(1+ζ). The principle of proofs rely on the p-Hilbert class field theory.