Class-determining field

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.

Facts

 * Non-modular implies class-determining: For a finite group, any field whose characteristic does not divide the order of the group is a class-determining field.
 * Cyclic implies every field is class-determining
 * Elementary abelian of prime-square order implies corresponding prime field is not class-determining