Minimal splitting field need not be contained in a cyclotomic extension of rationals
Contents
Statement
It is possible to have a finite group and a minimal splitting field
of
such that
has characteristic zero, but it is not contained in any cyclotomic extension of the rationals.
Note that, by the Kronecker-Weber theorem from Galois theory, being contained in a cyclotomic extension of the rationals is equivalent to being an abelian extension of the rationals, so this is equivalent to saying that it is possible for a finite group to have a minimal splitting field that is not an abelian extension of the rationals.
Related facts
Similar facts
- Minimal splitting field need not be unique
- Splitting not implies sufficiently large
- Field generated by character values is splitting field implies it is the unique minimal splitting field
- Minimal splitting field need not be cyclotomic
Opposite facts
- Sufficiently large implies splitting: This in particular shows that there exists at least one minimal splitting field that is contained in a cyclotomic extension of rationals.
- Field generated by character values is contained in a cyclotomic extension of rationals: Combined with field generated by character values is splitting field implies it is the unique minimal splitting field (a situation that occurs when all irreducible representations have Schur index one), it shows that to find an example of a minimal splitting field that is not contained in a cyclotomic extension of the rationals, we need to look for a finite group whose field generated by character values is not a splitting field.
Proof
Algebraic example
Further information: linear representation theory of quaternion group, faithful irreducible representation of quaternion group
Consider the quaternion group. Apart from the two-dimensional faithful irreducible representation of quaternion group, all other representations can be realized over the rationals, so a field in characteristic zero is a splitting field iff the two-dimensional faithful irreducible representation can be realized over it.
A sufficient condition for being a splitting field is that the field contain elements such that
. We construct one such field and then argue that no proper subfield of it is a splitting field.
Field description
Consider the field where where
is the real cube root of 2 and
is a square root of
.
Proof that this field is a minimal splitting field for the quaternion group
We have . The extension degree
is 3 and the extension degree
is 2 (note that the latter extension is strict because
can be viewed as a subfield of
but
cannot). Overall, the extension
has degree
.
As noted above, the field is a splitting field for the quaternion group.
Thus, it suffices to show that it contains no proper subfield that is also a splitting field for the quaternion group. For this, note that any proper subfield which is degree three over the rationals cannot work as it is formally real, and there are no proper subfields that are degree two over the rationals.
Proof that the field is not contained in a cyclotomic extension of the rationals
Any cyclotomic extension of the rationals has an abelian Galois group, so any subfield of it must also be a Galois extension. However, is not Galois, because the automorphism mapping
to a complex cube root of 2 does not preserve the subfield. Therefore
is also not contained in a cyclotomic extension of the rationals.
For this part of the proof, see also this Math StackExchange question.