Galois Module Structure of \Z/\ell^n-th Classes of Fields
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.
Cite
@article{arxiv.1204.6611,
title = {Galois Module Structure of \Z/\ell^n-th Classes of Fields},
author = {Jan Minac and John Swallow and Adam Topaz},
journal= {arXiv preprint arXiv:1204.6611},
year = {2015}
}
Comments
Final version: to appear in Bull. of the London Math Soc