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CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passverified 2026-09-23 (gpt-6-sol)
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The Schur index divides the representation degree

Statement

Assume the Axiom of Choice. Let χ be an irreducible complex character, put K=Q(χ), and let V be the irreducible K-representation used in The Schur index of an irreducible character. Then mK(χ)∣dim⁡KV.

Facts & Assumptions

Given: AC, χ, K, and V as in the statement, together with a finite Galois splitting field E/K inside C.

[L1]

dim⁡KV=mK(χ) [K(χ):K],χ(1) (Character formula over a nonsplitting field).

Proof

technique · direct
1.1L1givenalgebra

Under AC the attached V and mK(χ) exist by the definition. Since K(χ)=K, [L1] gives dim⁡KV=mK(χ)χ(1). The integer χ(1) is positive, so this expresses dim⁡KV as an integer multiple of mK(χ).

2.1step 1.1∎

Hence mK(χ)∣dim⁡KV.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources