noethers | MathBB

See the previous post for the notations and/or definitions used here without reference.

In 1915 Fischer [Fi15] proved that for ${G}$ (finite) abelian the extension ${k(G):=k(x_g\>|\> g\in G)^G}$ is always rational if ${k}$ contains “enough” roots of unity. Let us give the precise and general formulation first.

Theorem 1 (Fischer, [Fi15])

Let ${G}$ be an abelian group with exponent ${e}$ . Assume that ${k}$ is a field containing the ${e^{th}}$ roots of unity and either the characteristic of ${k}$ does not divide ${e}$ or is equal to 0. Then ${k(G)}$ is rational.

This implies, in particular, that ${{\mathbb C}(G)}$ is rational for all abelian ${G}$ !

For our choice of ${k}$ and ${G}$ — we have assumed that the field ${k}$ is of characteristic 0 and that ${G}$ is a finite cyclic group, see the previous post — Fischer’s theorem reduces to:

Theorem 2 (Fischer, special case)

Let ${G={\mathbf C}_n}$ and assume that the field ${k}$ (of characteristic 0) contains the ${n^{th}}$ roots of unity. Then ${k(G)}$ is rational.

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This blog post is meant to be the introduction of a series of posts about Noether’s Problem, which, roughly put, asks to study the subfield of a rational function field invariant under some automorphisms (namely to determine its rationality). Since there are quite a number of variants of Noether’s Problem, and also different methods of studying them, it seems necessary to give an outline of what I hope to achieve in these posts.

I studied Noether’s Problem as part of a project (for a course) in which my main goal was to try to understand the ideas behind results as Swan’s [Swa69] (counter)example and Lenstra’s [Len74] solution for (finite) abelian groups (in some special cases). Besides understanding these ideas and results, I also wanted to present them in the most straightforward way possible. This will also be the goal of this series of posts: giving an accessible introduction to studying Noether’s Problem for (finite) cyclic groups over ${{\mathbb Q}}$ where the results of Swan and Lenstra will be the main focus.

The prerequisites needed to be able to follow these posts should be relatively minimal. Basic group theory, Galois theory and some relatively basic facts about modules should essentially be all that is needed (which is almost all covered in the two algebra courses from my mathematics education up until now). Where I feel it might be useful and/or necessary (also for myself), I will expand on some of the more theoretic concepts or try to provide some references.

1. The Problem

Let ${k}$ be an arbitrary field. For an arbitrary finite group ${G}$ we can then define the rational function field ${k(x_g\>|\> g\in G)}$ , where the ${x_g}$ are just symbols or variables functioning as transcendent elements (over ${k}$ ). Hence, there is an obvious action of ${G}$ on ${k(x_g\>|\> g\in G)}$ : each ${h\in G}$ permutes the elements ${x_g}$ as ${h(x_g)=x_{gh}}$ . Noether’s Problem then asks the following question:

Noether’s Problem

When is the fixed field ${k(G):=k(x_g\>|\> g\in G)^G}$ rational (over k)?

Where by rational we mean purely transcendental, i.e., isomorphic to a rational function field over ${k}$ with the same transcendency degree ${|G|}$ . Thus, there exists a generating set ${\mathcal{B}}$ of algebraically independent elements such that ${k(\mathcal{B})=k(x_g\>|\> g\in G)}$ .