Sufficiently large implies character-separating

This fact is related to: linear representation theory
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Statement

Let $G$ be a finite group and $k$ a field whose characteristic does not divide the order of $G$. Then, if $k$ is a sufficiently large field for $G$ (that is, $k$ contains all the $m^{th}$ roots of unity where $m$ is the exponent of $G$), then $k$ is a character-separating field for $G$.

By $k$ is character-separating for $G$, we mean that given two distinct conjugacy classes $c_{1}$ and $c_{2}$ of $G$ and elements $g_{i}\in c_{i}$, there exists a linear representation $\rho$ whose character takes different values on $g_{1}$ and $g_{2}$.

Facts used

• Sufficiently large implies splitting: If $k$ is a sufficiently large field for $G$, then $k$ is a splitting field for $G$: every representation of $G$ over $k$ is completely reducible, and every representation irreducible over $k$ is irreducible over any field extension of $k$.
• Splitting implies character-separating: Any splitting field for a finite group is character-separating: given any two conjugacy classes, there is a linear representation whose character takes different values on these conjugacy classes.