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
Hence, we get the equivalent action
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.
[Fi15] Fischer, E.; Die Isomorphie der Invariantenkorper der endlichen Abelschen Gruppen linearen transformationen, 1915.
You must be logged in to post a comment.