Abstract
Let k be any field, G be a finite group. Theorem Assume that (i) G contains an abelian normal subgroup H so that G / H is cyclic of order n, (ii) Z [ ζ n ] is a unique factorization domain, and (iii) ζ e ∈ k where e is the exponent of G, i.e. e = lcm { ord ( g ) : g ∈ G } . If G → GL ( V ) is any finite-dimensional linear representation of G over k, then k ( V ) G is rational (= purely transcendental) over k.
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