Sufficiently large implies character-separating

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.