Minimal splitting field need not be unique
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.
- 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
- 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:
- 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 .