# Minimal splitting field need not be unique

## Contents

## Statement

### In characteristic zero

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

### In prime characteristic

Not sure whether there are examples here.

## Related facts

### Similar facts

- Splitting not implies sufficiently large
- Minimal splitting field need not be cyclotomic
- Minimal splitting field need not be contained in a cyclotomic extension of rationals

### Opposite facts

## 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:

- is not a splitting field.
- Any field of the form where is a splitting field.

Thus, any field of the form , where , is a quadratic extension of 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 and .