The Haase norm theorem mod squares
type: Item
instance of: scholarly article
title: The Haase norm theorem mod squares
zbMATH Open document ID: 0771.11045
DOI: 10.1016/0022-314X(92)90098-A
author: David B. Leep
author: A. R. Wadsworth
published in: Journal of Number Theory
publication date: 1993-01-17T00:00:00Z
review text: Let \(K/F\) be a Galois extension of global fields of finite degree. The Hasse norm principle for \(K/F\) is the statement that every element of \(F\) which is a norm in every local extension of \(K/F\) is also a (global) norm in \(K/F\) itself. It is well known that the Hasse norm principle holds for \(K/F\) if \(K/F\) is cyclic. In a previous paper the authors have considered the question under which conditions the Hasse norm principle modulo squares holds for \(K/F\) [see Trans. Am. Math. Soc. 315, 415-432 (1989; Zbl 0681.10015)]. Under the natural assumption \(\text{char}(F)\neq 2\) they have shown that the Hasse norm principle modulo squares is equivalent to a certain local-global principle in the theory of quadratic forms and that in case \(\text{Gal}(K/F)\cong(\mathbb{Z}/2)^ r\) the Hasse norm principle modulo squares holds, even if the usual Hasse norm principle does not hold. In the paper under review the authors consider the case \(\text{Gal}(K/F)\cong\mathbb{Z}/2\times \mathbb{Z}/2^ r\), \(r\geq 2\), and they show that the Hasse norm principle modulo squares holds if and only if the usual Hasse norm principle holds. As a corollary they obtain: If a square in \(F\) is a norm in every local extension of \(K/F\) then it is a norm in \(K/F\). The authors remark that for \(\text{Gal}(K/F)\) abelian of exponent \(p\), \(p\) an odd prime, the Hasse norm principle modulo \(p\)th powers holds if and only if the usual Hasse norm principle holds and that for \(\text{Gal}(K/F)\cong \mathbb{Z}/p^ r\times\mathbb{Z}/p\), \(r\geq 1\), the Hasse norm principle modulo \(p\)th powers is again equivalent to the usual Hasse norm principle.
reviewed by: Hans Opolka
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 98396
zbMATH Keywords: norms
zbMATH Keywords: abelian extensions of global fields
zbMATH Keywords: class field theory
zbMATH Keywords: Hasse norm principle
zbMATH Keywords: local-global principle
MaRDI profile type: Publication
cites work: Q5532660
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full work available at URL: 0022-314x(92)90098-a
OpenAlex ID: W2088330506
description: scientific article; zbMATH DE number 98396