Tag Archives: galois

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.

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