The Haase norm theorem mod squares
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.
scientific article; zbMATH DE number 98396
scientific article