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Noether’s problem for some meta-abelian groups of small degree. (English) Zbl 1083.12002

Let \(K= \mathbb{Q}(x_0,\dots, x_{n-1})\) be the field of rational functions in \(n\) variables and \(G\) a transitive subgroup of \(S_n\). Then \(G\) acts on \(K\) by permuting the variables \(x_0,\dots, x_{n-1}\). Emmy Noether asked whether the fixed field \(K^G\) is a purely transcendental extension of \(\mathbb{Q}\). If the question has an affirmative answer then \(G\) can be realized as a Galois group over \(\mathbb{Q}\) and there even exists a \(G\)-generic polynomial over \(\mathbb{Q}\). For some groups \(G\) the answer is affirmative, for other groups \(G\) the answer is negative, but so far the results here are of fragmentary.
The main result of this paper enlarges the list of groups for which it is known that Noether’s problem has an affirmative answer: Let \(\text{Aff}(\mathbb{Z}/n\mathbb{Z})\) be the group of affine one-dimensional transformations on \(\mathbb{Z}/n\mathbb{Z}\) (i.e. \(\text{Aff}(\mathbb{Z}/n\mathbb{Z})\rtimes(\mathbb{Z}/n\mathbb{Z})\times(\mathbb{Z}/n\mathbb{Z})^*)\). Then Noether’s problem has an affirmative answer for every subgroup \(G\) of \(\text{Aff}(\mathbb{Z}/n\mathbb{Z})\) containing \(\mathbb{Z}/n\mathbb{Z}\) for \(n= 9,10,12,14,15\).

MSC:

12F12 Inverse Galois theory
11R32 Galois theory
12F10 Separable extensions, Galois theory
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References:

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