Splitting not implies sufficiently large

Statement

A splitting field for a group need not be a sufficiently large field for the group.

Related facts

• Splitting field for a group not implies splitting field for every subgroup: In fact, this is an equivalent formulation, since sufficiently large implies splitting for every subquotient and splitting field for every subgroup implies sufficiently large.

Proof

Example of the symmetric groups

The symmetric groups on finite sets are rational-representation groups: the field of rational numbers is a splitting field for all of them. However, it is clearly not sufficiently large for symmetric groups of degree three or higher. The smallest example is symmetric group:S3, whose exponent is $6$. The rational numbers are a splitting field for this group but they do not contain the primitive sixth roots of unity.