# Splitting not implies sufficiently large

## Contents

## 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 . The rational numbers are a splitting field for this group but they do not contain the primitive sixth roots of unity.