Minimal splitting field need not be cyclotomic
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 group has a minimal splitting field that is contained in a suitable cyclotomic extension.
- 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
Examples where it is the unique minimal splitting field and is generated by character values
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|