# Sufficiently large implies character-separating

This fact is related to: linear representation theory

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## Statement

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

By is character-separating for , we mean that given two distinct conjugacy classes and of and elements , there exists a linear representation whose character takes different values on and .

## Facts used

- Sufficiently large implies splitting: If is a sufficiently large field for , then is a splitting field for : every representation of over is completely reducible, and every representation irreducible over is irreducible over any field extension of .
- 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.