Minimal splitting field need not be cyclotomic

Statement

In characteristic zero

It is possible to have a finite group $G$ and a minimal splitting field $K$ in characteristic zero that is not a cyclotomic extension of the rationals. Further, we can choose examples of both the following sorts:

• Examples where $K$ is the unique minimal splitting field for $G$, on account of being the field generated by character values.
• Examples where $G$ 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 $n$ and order $2n$ is $\mathbb {Q} (\cos(2\pi /n))$, which is a subfield of the reals. When $n\neq 1,2,3,4,6$, 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

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	$\mathbb {Q}$	$\mathbb {Q} ({\sqrt {-2}})$	$\mathbb {Q} ({\sqrt {-1}})$	linear representation theory of quaternion group	faithful irreducible representation of quaternion group