Minimal splitting field

Definition

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

Facts

Uniqueness and relation with field generated by character values

• The minimal splitting field in any characteristic must contain the field generated by character values in that characteristic.
• Field generated by character values is splitting field implies it is the unique minimal splitting field. If all representations have Schur index 1, then these equivalent conditions hold.
• 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 (and the unique minimal splitting field in that characteristic), because prime fields have no proper subfields of any sort.
• 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)

Relation with sufficiently large fields and cyclotomic fields

• Sufficiently large implies splitting: Any sufficiently large field, i.e., any field that contains that $d^{th}$ primitive roots of unity where $d$ 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 cyclotomic
• Minimal splitting field need not be contained in a cyclotomic extension of rationals