= Splitting field for finite group representations
A <field> $k$ is a splitting <field> for a finite <group> $G$ if every <simple module> over its <group algebra> $kG$ is absolutely simple: it remains simple over every extension <field>. Equivalently, the quotient by the <Jacobson radical> is a product of full <matrix algebras> over $k$. In <characteristic> $p$, containing all $m$th <roots of unity>, where $|G|=p^a m$ and $p\nmid m$, is sufficient. This is the modular splitting-field theorem; it is substantially stronger than merely having <eigenvalues> for one chosen group element.
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