Abstract
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∈G} . Let F( G) G denote the fixed field of F( G) under the action of G. Let w be a primitive ( p−1)st root of 1, and let I be the ideal ( p, w− a) in Z[ w] where a is a primitive ( p−1)st root of 1mod 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.
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