Splitting not implies sufficiently large

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Statement

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

Related facts

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.