DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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A splitting field for a finite group: every irreducible representation has scalar endomorphism ring

Definition

Let $G$ be a finite group and let $k$ be a field. The field $k$ is a splitting field for $G$ when every irreducible representation $V$ of $G$ over $k$ satisfies ${End}_{G} (V) = k,$ meaning that every $G$ -endomorphism of $V$ is a scalar operator $λ {id}_{V}$ with $λ \in k$ (A finite-dimensional representation $ρ : G \to GL (V)$ over a field, and its degree, Intertwiners, the spaces ${Hom}_{G} (V, W)$ and ${End}_{G} (V)$ , equivalent representations, and faithful representations).

Remarks

This is the representation-theoretic condition used on the next pages. It is weaker than algebraic closedness, and Brauer's theorem records one sufficient roots-of-unity criterion for it without making that criterion the definition.
The definition quantifies only over irreducible representations of the fixed finite group $G$ .

Depends on

Used by

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Sources

Peter Webb, A Course in Finite Group Representation Theory, Chapter 9 Sections 9.1-9.2 (standard reference, not scraped)