Sur la pro-p-extension localement cyclotomique maximale d'un corps de nombres

TL;DR

This paper investigates the structure of a specific Galois group over a number field's cyclotomic extension, linking its freeness to Galois descent and providing criteria to determine non-freeness.

Contribution

It establishes a connection between the freeness of the Galois group and Galois descent, offering explicit criteria to identify when the group is not free.

Findings

Identifies conditions under which the Galois group is not free pro-p.

Links the freeness of the Galois group to Galois descent properties.

Provides explicit criteria for non-freeness of the Galois group.

Abstract

Let p be a prime number and F be a number field. We consider the Galois group G over the cyclotomic Z_p extension of F of the maximal unramified, p-decomposed, pro-p-extension of the cyclotomic Z_p extension of F. The question whether G is free pro-p was already asked by many authors. In this article, we highlight a link between the freeness of G and the Galois descent for some localisation kernels. Then we give explicit criterions to show that G is not a free pro-p-group.