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

In 1915 Fischer [Fi15] proved that for (finite) abelian the extension is always rational if contains “enough” roots of unity. Let us give the precise and general formulation first.

Theorem 1 (Fischer, [Fi15])

Let be an abelian group with exponent . Assume that is a field containing the roots of unity and either the characteristic of does not divide or is equal to 0. Then is rational.

This implies, in particular, that **is rational for all abelian** !

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

Theorem 2 (Fischer, special case)

Let and assume that the field (of characteristic0) contains the roots of unity. Then is rational.