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 characteristic 0) contains the roots of unity. Then is rational.