Galois module structure of \(p\)th-power classes of extensions of degree \(p\)
type: Item
instance of: scholarly article
title: Galois module structure of \(p\)th-power classes of extensions of degree \(p\)
zbMATH Open document ID: 1040.12006
DOI: 10.1007/BF02783417
published in: Israel Journal of Mathematics
publication date: 2004-03-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0202224
review text: Let \(p\) be a prime number (possibly 2). Let \(F\) be a field with characteristic not \(p\) and containing a primitive \(p\)th root of unity. For all cyclic extensions \(K/F\) of degree \(p\), the authors determine the Galois module structure of the group \(J:= K^\times/K^{\times p}\) of \(p\)th power classes of \(K\) (where \(K^\times= K\setminus\{0\}\)). To know the Galois module structure of \(J\) is a fundamental question related to both the structure of the Galois cohomology of \(F\) and the structure of the maximal \(p\)-extension of \(F\). Just as in the case of local fields, the structure depends upon only the norm subgroup, its intersection with the subgroup of \(p\)th roots of unity, and the multiplicative subgroup of the base field. The proofs require only Hilbert's Theorem 90, basic Kummer theory and linear algebra over the group ring \(\mathbb F_p[\text{Gal}(K/F)]\).
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 2060051
zbMATH Keywords: \(p\)th-power classes
author: Ján Mináč
author: John R. Swallow
reviewed by: Richard Massy
MaRDI profile type: Publication
arXiv ID: math/0202224
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description: scientific article