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
.