Characters span class functions iff they separate conjugacy classes iff field contains field generated by character values

Statement
Suppose $$G$$ is a group and $$k$$ is a field whose characteristic does not divide the order of $$G$$. The following are equivalent:


 * 1) The characters of irreducible representations of $$G$$ over $$k$$ form a basis for the space of class functions.
 * 2) The characters of irreducible representations of $$G$$ over $$k$$ span the space of class functions.
 * 3) The characters of all finite-dimensional linear representations of $$G$$ over $$k$$ span the space of class functions.
 * 4) Given any two distinct conjugacy classes of $$G$$, there is an irreducible representation whose character value is different on the two conjugacy classes.
 * 5) $$k$$ contains a subfield $$K$$ that is isomorphic to the field generated by character values in its characteristic, i.e., there is a splitting field containing $$k$$ such that all irreducible representations of $$G$$ over the splitting field have character values in $$k$$.

Note that $$k$$ need not itself be a splitting field for $$G$$. For instance, the field of rational numbers satisfies these equivalent conditions for the quaternion group (see linear representation theory of quaternion group, faithful irreducible representation of quaternion group).

Facts used

 * 1) uses::Splitting implies characters form a basis for space of class functions, uses::splitting implies characters separate conjugacy classes