# Field generated by character values need not be cyclotomic

From Groupprops

## Contents

## Statement

It is possible to have a finite group such that the field generated by character values in characteristic zero is not a cyclotomic extension of the field of rational numbers.

## Related facts

### Similar facts

- Minimal splitting field need not be cyclotomic
- Minimal splitting field need not be contained in a cyclotomic extension of rationals

### Opposite 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

## Proof

### Examples where the field generated by character values is the unique minimal splitting field

`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 coincides with the field generated by character values. It 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 |