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.

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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 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