---
tag: JEMV9NEDGR
kind: concept
name: Defined over
sources:
  - wikipedia: en:55799101@r1351912484
    locator: Glossary of representation theory, section D
    note: adopted
    mention: wikipedia://en/55799101#ZLC2
    sections: [Summary]
inventory: [A9YYNA7ARA]
---

## Summary

Given a field extension $K/F$, a representation *V* of a group *G* over *K* is said to be defined over *F* if $V \simeq V_0 \otimes_F K$ for some representation $V_0$ over *F* such that $g: V \to V$ is induced by $g: V_0 \to V_0$; i.e., $g \cdot (v \otimes \lambda) = g v \otimes \lambda$. Here, $V_0$ is called an *F*-form of *V* (and is not necessarily unique).