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

Abstract

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.