TheoremStatement: 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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Character formula over a nonsplitting field

Statement

Let $F \subseteq E \subseteq C$ , let $V$ be an irreducible $F$ -representation of a finite group $G$ , and let $E / F$ be finite Galois and a splitting field for $G$ . Let $χ$ be the character of one absolutely irreducible constituent of $E \otimes_{F} V$ , let $m$ be that constituent's multiplicity in $E \otimes_{F} V$ , and put $H = Stab (χ)$ . Then $χ_{V} = m \sum_{σ \in Gal (E / F) / H} σ χ .$ In particular, $\dim_{F} V = m [F (χ) : F] χ (1)$ .

Facts & Assumptions

Given: $F \subseteq E \subseteq C$ , $G$ , $V$ , $χ$ , $m$ , and $H$ as in the statement.

[L1]

Scalar extension of $V$ is $m$ times the orbit of $χ$ 's representation (Scalar extension of an irreducible finite-group representation).

[L2]

The character of a direct sum is the sum of the characters (Characters add on direct sums, multiply on tensor products, and conjugate on duals).

[L3]

The fixed field of the stabilizer is $F (χ)$ (The character field is the stabilizer fixed field).

Proof

technique · direct

1.1L1L2

By [L1] and [L2], taking characters of the scalar-extension decomposition gives the displayed character identity.

2.1step 1.1algebra

Evaluating that identity at $1_{G}$ gives $\dim_{F} V = m [Gal (E / F) : H] χ (1)$ .

3.1L3step 2.1∎

By [L3] and the finite Galois correspondence, the index $[Gal (E / F) : H]$ equals $[F (χ) : F]$ . Substitute this into step 2.1.

Depends on

Used by

Dependency tree · two levels

18 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

Weizhe Zheng, Lectures on Algebra, Corollary 4.3.3 (standard reference, not scraped)
Gabor Wiese, Galois Representations, Corollary 2.5.4 (standard reference, not scraped)