# Minimal splitting field

From Groupprops

## Definition

Let be a finite group and be a field whose characteristic does not divide the order of (so may have characteristic zero or some prime coprime to the order of ). We say that is a **minimal splitting field** for if is a splitting field for and no proper subfield of is a splitting field for .

## Facts

- If a prime field (i.e., either the rationals or a field of prime size) is a splitting field, then it is a minimal splitting field, because prime fields have no proper subfields of any sort.
- Sufficiently large implies splitting: Any sufficiently large field, i.e., any field that contains that primitive roots of unity where is the exponent of the group, is a splitting field. In particular, this means that every sufficiently large field contains a minimal splitting field.
- Splitting not implies sufficiently large, and the minimal sufficiently large field need not be a minimal splitting field.
- Minimal splitting field need not be unique: It is possible for a group to have multiple non-isomorphic minimal splitting fields. For this to occur, at least one of the irreducible representations must have Schur index greater than 1. An example is the quaternion group (see linear representation theory of quaternion group). On the other hand, if the field generated by the character values is a splitting field, then it is the unique minimal splitting field.
- Minimal splitting field need not be cyclotomic