Minimal splitting field need not be unique

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 $$L$$.

In prime characteristic
Not sure whether there are examples here.

Similar facts

 * Splitting not implies sufficiently large
 * Minimal splitting field need not be cyclotomic

Opposite facts

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

Example of the 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 $$\mathbb{Q}(\sqrt{-2}) = \mathbb{Q}[t]/(t^2 + 2)$$.