2006/10/31 by Roland Queme, Queme, Roland
Mathematics · #11R18 #11R29 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R18 #msc:11R29
paper · pdf · doi:10.48550/arxiv.math/0610968
A correction in proof of lemma 2.2 p. 3 is made
arxiv created 2006/11/12 · arxiv updated 2009/12/01
Let p be an odd prime. Let Fp^* be the no-null part of the finite field of p elements. Let K = Q(zeta) be the p-cyclotomic field and let OK be the ring of integers of K. Let pi be the prime ideal of K lying over p. An integer B ∈ OK is said singular if B1/p not ∈ K and if B OK = bp where b is an ideal of OK. An integer B ∈ OK is said semi-primary if B = beta mod pi2 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 ∈ Fp^*. We prove the existence, for each Gamma, of singular semi-primary integers B where B OK= bp with class Cl(b) ∈ Gamma and Bsigma-mu ∈ Kp and we describe their pi-adic expansion. This paper is at a strictly elementary level.