Galois Module Structure of \Z/\ell^n-th Classes of Fields

TL;DR

This paper investigates the Galois module structure of certain field class groups using the Merkurjev-Suslin theorem, providing new conditions for submodule embeddings and generalizing previous results for specific cases.

Contribution

It introduces a precise criterion, in terms of cohomology, for submodule embeddings in Galois modules arising from Kummer theory, extending known theorems to broader settings.

Findings

Derived conditions for submodule embeddings in Galois modules

Generalized a theorem of Adem, Gao, Karaguezian, and Minac

Provided a cohomological description analogous to Hilbert's Theorem 90

Abstract

In this paper we use the Merkurjev-Suslin theorem to explore the structure of arithmetically significant Galois modules that arise from Kummer theory. Let K be a field of characteristic different from a prime \ell, n a positive integer, and suppose that K contains the (\ell^n)^th roots of unity. Let L be the maximal \Z/\ell^n-elementary abelian extension of K, and set G = \Gal(L|K). We consider the G-module J = L^\times/\ell^n and denote its socle series by J_m. We provide a precise condition, in terms of a map to H^3(G,\Z/\ell^n), determining which submodules of J_{m-1} embed in cyclic modules generated by elements of J_m. This generalizes a theorem of Adem, Gao, Karaguezian, and Minac which deals with the case m=\ell^n=2. This description of J_m/J_{m-1} can be viewed as an analogue of the classical Hilbert's Theorem 90 and it is helpful for understanding the G-module J.