Splitting field for a group not implies splitting field for every subgroup

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Statement

A splitting field for a finite group need not be a splitting field for every subgroup.

Related facts

Proof

Examples

Further information: Linear representation theory of symmetric group:S3, linear representation theory of symmetric groups

Consider symmetric group:S3, the symmetric group of degree three. The rational numbers are a splitting field for this group. However, they are not a splitting field for the alternating group of degree three, because there exists a character of this group that is not realizable over the rationals.

More generally, by Cayley's theorem, any finite group can be embedded as a subgroup in a symmetric group of finite degree, and symmetric groups are rational-representation groups, i.e., the rational numbers are a splitting field for every symmetric group. On the other hand, the rational numbers are not a splitting field for most finite groups, giving a plethora of counterexamples.