# 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 |