LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passverified 2026-09-23 (gpt-6-sol)

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The Schur index is independent of the splitting field

Statement

Assume the Axiom of Choice. For an irreducible character $χ$ of a finite group, the common multiplicity in the scalar-extension orbit used in The Schur index of an irreducible character is unchanged when the finite Galois splitting field is replaced by a larger finite Galois splitting field.

Facts & Assumptions

Given: AC, $K = Q (χ)$ , the irreducible $K$ -module $V$ attached to $χ$ by the definition, and finite Galois splitting fields $E \subseteq L$ over $K$ .

[L1]

Scalar extension is associative: $L \otimes_{K} V ≅ L \otimes_{E} (E \otimes_{K} V)$ (Change of rings: $N \otimes_{R} M ≅ N \otimes_{S} (S \otimes_{R} M)$ ).

[L2]

Over either splitting field, an irreducible $K$ -module extends as one Galois orbit with a common multiplicity (Scalar extension of an irreducible finite-group representation).

Proof

technique · direct

1.1L2given

Under AC, the attached $V$ exists by The Schur index of an irreducible character. Write $E \otimes_{K} V$ as $m$ times its orbit of pairwise inequivalent absolutely irreducible constituents, using [L2].

2.1L1step 1.1

Each constituent remains irreducible after extension from $E$ to the splitting field $L$ , and distinct constituents remain distinct; therefore [L1] writes $L \otimes_{K} V$ with the same coefficient $m$ .

3.1L2step 2.1∎

Applying [L2] directly over $L$ identifies its common multiplicity with that coefficient. Hence both choices give $m$ , proving independence.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gabor Wiese, Galois Representations, Proposition 2.3.13 and Corollary 2.5.4 (standard reference, not scraped)