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.
This is an important and interesting theorem. Not only does it give us some idea of how we can handle rationality, its proof is also relatively short and elementary. Furthermore, the concepts introduced in the proof will be used throughout the series.
Proof. Let denote the standard basis for the -dimensional vector space over . In we can look at the “linear action” of on , simply by making the correspondence . Since acts on by (by definition), the action of on corresponds with a cyclic permutation matrix permutating the basis vectors .
As contains the roots of unity, the permutation matrix is diagonalizable with linearly independent eigenvectors () and eigenvalues the roots of unity: (where denotes a primitive root of unity). These eigenvectors correspond in with algebraically independent elements which we can define explicitely as
of on .
Then I claim that is a basis for the extension (and hence that is rational).
The are fixed by (quick check) — and thus by — such that we have the inclusion . As is the Galois group of , we have that . It is easy to see that , but — and now comes the essential part — is a root of the polynomial , thus . Or in a diagram:
We conclude and the claim is proved.
First of all, note that not only did we prove that is rational, we also have an explicit basis , defined in Eq. 3, for it! Secondly, the proof could be made somewhat shorter by just giving the basis and checking that it is indeed a basis for , as in the last paragraph. Personally, I think this longer version provides some meaningful context.
To conclude, let us make the proof of Fischer’s theorem a little more explicit by considering an example.
Example 1 ()
Let and . As in the previous post, works on by definition of . The cyclic permutation matrix defined in the proof is given by
which acts on the standard basis corresponding with the algebraically independent elements . Such that, for example, we have the correspondence
The eigenvectors of , satisfying , are given by
From the eigenvectors we can read off the corresponding algebraically independent elements
and we can easily check that they indeed satisfy . For example,
Finally, for computational reasons we replace the basis , defined in general as Eq. 3, by the equivalent basis . (We have replaced by .) We get
with which we have .
In the next post we will use Fischer’s theorem to formulate the basic idea on which we will build our “complete solution” for and ( prime). We will also prove an interesting theorem from Masuda that will allow us to prove the rationality for some small order groups.