Minimal splitting field need not be cyclotomic

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

Examples where it is the unique minimal splitting field and is generated by character values
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 $$n \ne 1,2,3,4,6$$, 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):