Abstract
Let K be any field and G be a finite group. Let G act on the rational function field by K-automorphisms defined by for any . Denote by the fixed field . Noether's problem asks whether is rational (i.e. purely transcendental) over K. We will prove that, if K is any field, p an odd prime number, and G is a nonabelian group of exponent p with or p 4 satisfying , then is rational over K. Moreover, it will be shown that is retract rational if G belongs to a much larger class of p-groups. In particular, generic G-polynomials of G-Galois extensions exist for these groups.
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