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- W2076670666 abstract "Let G be a finite group having a cyclic quotient of order 8. Let L be the field given by adjoining indeterminates {xg g E G} to Q. Then LG is not a rational function field. Let G be a finite group, {xg: g E G} a set of commuting indeterminates and L the rational function field over Q in the xg. Then G acts as a group of automorphisms of L by the rule xh = Xgh, g, h E G. The Noether problem is whether or not LG is a rational function field. This question was first given a negative answer by Swan [3] for certain cyclic groups G, the smallest being of order 47. For abelian groups the question was solved completely by Lenstra [1] and recently Saltman [2] has shed a new light on the question via his theory of generic Galois extensions. Lenstra classified all abelian G for which LG is rational and in particular obtained a smallest negative example: LG is not rational for G a cyclic group of order 8. Saltman obtains the nonrationality of LG for this group as well as for any abelian group having an element of order 8 [2, Theorem 5.11]. There do not appear to be examples of nonrational LG for nonabelian G in the literature. The purpose of this note is to make the observation that Saltman's results generate nonabelian examples as well as abelian ones determined earlier by Lenstra. We begin by pointing out that the proof of Saltman's theorem [2, Theorem 5.11] on the nonrationality of LG for abelian groups having an element of order 8 actually applies to any split extension" @default.
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- W2076670666 date "1985-02-01" @default.
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- W2076670666 title "Nonabelian counterexamples to the Noether problem" @default.
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