Stable rationality of certain invariant fields

[摘要] Let F be a field. For a finite group G, let F(G) be the purely transcendental extension of F with transcendency basis {x(g): g is an element of G}. Let F(G)(G) denote the fixed field of F(G) under the action of G. Let omega be a primitive (p - 1)st root of 1, and let I be the ideal (p, omega - a) in Z[omega] where a is a primitive (p - I)st root of I mod p. We show that if G be the semi-direct product of a cyclic group of order p by a cyclic group of order prime to p, if I is principal, and if F contains a primitive \G\th root of 1, then F(G)(G) is stably rational over F. It is not known whether the set of primes p for which I is principal is finite or infinite. We also show that if p is an odd prime and G is a non-abelian group of order p(3), then F(G)(G) is stably rational over F provided that F contains a primitive \G\th root of 1. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-11-15

[关键词] Noether settings;stable rationality;invariant fields