Minimal splitting field need not be unique

Statement

In characteristic zero

Let $G$ be a finite group. It is possible for $G$ to have two distinct non-isomorphic minimal splitting fields $K$ and $L$ in characteristic zero. In other words, both $K$ and $L$ are splitting fields, no proper subfield of either is a splitting field, and $K$ is not isomorphic to $L$.

Related facts

• Field generated by character values is splitting field implies it is the unique minimal splitting field

Proof

Example of the quaternion group

Further information: quaternion group, linear representation theory of quaternion group, faithful irreducible representation of quaternion group

The quaternion group of order eight has many different minimal splitting fields in characteristic zero. Specifically the following are true:

• $\mathbb{Q}$ is not a splitting field.
• Any field of the form $\mathbb{Q}(\alpha ,\beta )$ where ${\alpha }^2+{\beta }^2=-1$ is a splitting field.

Thus, any field of the form $\mathbb{Q}(\sqrt{-m^2-1})=\mathbb{Q}[t]/(t^2+m^2+1)$, where $m\in \mathbb{Q}$, is a quadratic extension of $\mathbb{Q}$ satisfying the condition for being a splitting field, and hence is a minimal splitting field. There are multiple non-isomorphic fields of this type, such as $\mathbb{Q}(i)=\mathbb{Q}[t]/(t^2+1)$ and $\mathbb{Q}(\sqrt{-2})=\mathbb{Q}[t]/(t^2+2)$.