# Class-determining field

This term is related to: linear representation theory
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This term associates to every group, a corresponding field property. In other words, given a field, every field either has the property with respect to that group or does not have the property with respect to that group

## Definition

### Symbol-free definition

A field is said to be a class-determining field for a group if any finite-dimensional linear representation is determined by the conjugacy classes in which the images of the conjugacy classes of the group lie under that representation.

In other words, for any two distinct (i.e. inequivalent) linear representations, there exists a conjugacy class whose image under the two representations does not lie in the same conjugacy class in the general linear group.

Equivalently, no two inequivalent linear representations are locally conjugate.

### Definition with symbols

A field $k$ is termed a class-determining field for a group $G$ if for any two inequivalent finite-dimensional linear representations $\rho_1,\rho_2:G \to GL(V)$, there exists $g \in G$ such that $\rho_1(g)$ and $\rho_2(g)$ are not conjugate.