# Sufficiently large implies character-separating

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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 $g_2$.
• 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 $k$.