Minimal splitting field need not be cyclotomic
Contents
Statement
In characteristic zero
It is possible to have a finite group and a minimal splitting field
in characteristic zero that is not a cyclotomic extension of the rationals. Further, we can choose examples of both the following sorts:
- Examples where
is the unique minimal splitting field for
, on account of being the field generated by character values.
- Examples where
has another minimal splitting field that is cyclotomic.
Note, however, that since sufficiently large implies splitting, any finite group has a minimal splitting field that is contained in a cyclotomic extension of the rationals.
Related facts
- Minimal splitting field need not be unique
- Sufficiently large implies splitting
- 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 contained in a cyclotomic extension of rationals
Proof
Examples where it is the unique minimal splitting field and is generated by character values
Further information: linear representation theory of dihedral groups, dihedral group:D16, linear representation theory of dihedral group:D16, faithful irreducible representation of dihedral group:D16
There are many examples among dihedral groups. The minimal splitting field for a dihedral group of degree and order
is
, which is a subfield of the reals. When
, then this is strictly bigger than
, and hence is not a cyclotomic extension of
.
Here are some examples (including dihedral groups and others):
Group | Minimal splitting field = Field generated by character values | Information on linear representation theory | Information on a faithful irreducible representation that requires use of the extension and cannot be realized over ![]() |
---|---|---|---|
dihedral group:D10 | ![]() |
linear representation theory of dihedral group:D10 | faithful irreducible representation of dihedral group:D10 |
dihedral group:D16 | ![]() |
linear representation theory of dihedral group:D16 | faithful irreducible representation of dihedral group:D16 |
semidihedral group:SD16 | ![]() |
linear representation theory of semidihedral group:SD16 | faithful irreducible representation of semidihedral group:SD16 |
Examples where there are other minimal splitting fields that are cyclotomic
Group | Field generated by character values | Minimal splitting field that is not cyclotomic | Minimal splitting field that is cyclotomic | Information on linear representation theory | Information on a faithful irreducible representation that requires use of either of the extensions |
---|---|---|---|---|---|
quaternion group | ![]() |
![]() |
![]() |
linear representation theory of quaternion group | faithful irreducible representation of quaternion group |