DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07

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Galois conjugates of a representation

Definition

Let $E / F$ be finite Galois, let $G$ be a group, and let $V$ be an $E$ -representation with matrices $ρ (g)$ in an $E$ -basis. For $σ \in Gal (E / F)$ , the $σ$ -conjugate $^{σ} V$ is the $E$ -representation whose matrix for $g$ is obtained by applying $σ$ entrywise to $ρ (g)$ . This does not depend on the chosen basis up to equivalence: conjugating every entry of $A ρ (g) A^{- 1}$ gives $σ (A) σ (ρ (g)) σ (A)^{- 1}$ .

Its character is $χ_{^{σ} V} = σ \circ χ_{V}$ . We write $Stab (V)$ for the subgroup of Galois automorphisms for which $^{σ} V ≅ V$ .

Depends on

Used by

Dependency tree · two levels

15 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, Definition 2.2.7 (standard reference, not scraped)
Weizhe Zheng, Lectures on Algebra, Proposition 4.6.14 (standard reference, not scraped)