Geometric generalization of Gaussian period relations with application in Noether’s problem for meta-cyclic groups. (English) Zbl 1081.12002

Hashimoto, Ki-ichiro; Hoshi, Akinari

Tokyo J. Math. 28, No. 1, 13-32 (2005).

This article studies Noether’s problem over \(\mathbb{Q}\) for metacylic groups. Let \(K=\mathbb{Q}(x_1, \dots, x_n)\) be the field of rational functions in \( n\) variables on which the symmetric group \(S_n\) of degree \(n\) acts through the permutation of the variables, and let \(G\) be a transitive subgroup of \(S_n\). Noether’s problem for \(G\) over \(\mathbb{Q}\) asks whether the fixed field \(K^G\) of \(G\)-invariant elements of \(K\) is again a rational function field over \(\mathbb{Q}\). In the case when it is rational, one also wants to obtain an explicit construction a set of independent generators of \(K^G\) over \(\mathbb{Q}\), from which one can obtain a \(\mathbb{Q}\)-generic \(G\)-polynomial. Noether’s problem has been studied by a number of authors in the case of abelian groups, but little is known for nonabelian groups beyond the classical case of \(G = S_n\) where the fixed field is generated by the symmetric polynomials. In this article it is shown that Noether’s problem has an affirmative answer for dihedral groups of order \(2n, n \leq 6\) and for the Frobenius group \(F_{20}\) of order 20. The solution given provides explicit constructions of independent generators of the fixed fields. The method employed is described by the authors as a geometric generalization of Gaussian period relations, working on a case-by-case basis and employing direct computation with the assistance of MAPLE and Mathematica.

Reviewer: Tara L. Smith (Cincinnati)

MSC:

12F12	Inverse Galois theory
12F10	Separable extensions, Galois theory

Keywords:

Noether’s problem; metacyclic groups

Software:

Maple; Mathematica

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