# Field generated by character values need not be cyclotomic

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## Statement

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

## Proof

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

There are many examples among dihedral groups. The minimal splitting field for a dihedral group of degree $n$ and order $2n$ coincides with the field generated by character values. It is $\mathbb{Q}(\cos(2\pi/n))$, which is a subfield of the reals. When $n \ne 1,2,3,4,6$, then this is strictly bigger than $\mathbb{Q}$, and hence is not a cyclotomic extension of $\mathbb{Q}$.

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 $\mathbb{Q}$
dihedral group:D10 $\mathbb{Q}(\cos(2\pi/5)) = \mathbb{Q}(\sqrt{5})$ linear representation theory of dihedral group:D10 faithful irreducible representation of dihedral group:D10
dihedral group:D16 $\mathbb{Q}(\cos(\pi/4)) = \mathbb{Q}(\sqrt{2})$ linear representation theory of dihedral group:D16 faithful irreducible representation of dihedral group:D16
semidihedral group:SD16 $\mathbb{Q}(\sqrt{-2})$ linear representation theory of semidihedral group:SD16 faithful irreducible representation of semidihedral group:SD16