# Tag Archives: fischer

## Noether’s Problem – 2. Fischer

##### 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.