Nonexcellent finite field extensions of 2-primary degree
type: Item
instance of: scholarly article
title: Nonexcellent finite field extensions of 2-primary degree
zbMATH Open document ID: 1358.11055
author: A. S. Sivatski
published in: Manuscripta Mathematica
publication date: 2016-12-20T00:00:00Z
review text: Let \(F\) be a field of characteristic different from \(2\). A field extension \(L/F\) is called excellent if for any quadratic form \(\varphi\) over \(F\), the anisotropic part of the form obtained by extending scalars to \(L\) is defined over \(F\). The extension \(L/F\) is called strongly excellent if for any extension \(K/F\) linearly disjoint with \(L/F\), the extension \(KL/F\) is excellent. For example, odd degree and quadratic extensions are well known to be strongly excellent. This paper considers the question as to whether the only strongly excellent extensions are those that can be realised as a quadratic extension followed by an odd degree extension. It is shown that any field extension \(L/F\) of degree \(2^m\) with \(m\geq 2\) is not strongly excellent. This is done via the following construction. First, it may be assumed that \(F\) has no finite odd degree extensions and that there exists an intermediate field \(F\subset k \subset L\) with \(k=F(\sqrt{u+v\sqrt{a}})\) and such that \(k/F\) is Galois. The proof is then divided into cases based on the Galois group \(G\) of \(k/F\), with the case \(G=\mathbb{Z}/4\mathbb{Z}\) being the most difficult. In this case, the fields \(F\) and \(L\) are replaced by \(F(x)\) and \(L(x)\) with \(x\) an indeterminate. A field extension \(K/F\) and a quaternion algebra \(D\) are then constructed such that \[ D\in {_2\mathrm{Br}}(K(\sqrt{x}, \sqrt{u+v\sqrt{a}})/K) \quad\mathrm{ and }\quad D\notin {_2\mathrm{Br}}(K(\sqrt{x})/K) + {_2\mathrm{Br}}(LK/K)\,. \] A quadratic form may be associated with this quaternion algebra, and the above conditions show that this quadratic form becomes isotropic over \(L\), but its anisotropic part over \(L\) is not defined over \(F\). The second of these two conditions on \(D\) is the more difficult, and requires computations with \(D\) in the cohomology groups \(H^1(K,\mathbb{Z}/2\mathbb{Z})\) and \(H^3(K,\mathbb{Z}/2\mathbb{Z})\). The results required for the construction of \(D\) are a…
reviewed by: Andrew Dolphin
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 12E20
zbMATH DE Number: 6666258
zbMATH Keywords: anisotropic form
zbMATH Keywords: nonexcellent extensions
zbMATH Keywords: Brauer group
MaRDI profile type: Publication
full work available at URL: s00229-016-0851-1
OpenAlex ID: W2408089328
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DOI: 10.1007/S00229-016-0851-1
description: scientific article; zbMATH DE number 6666258