Rational character (awaiting review)

A rational character of a finite group $G$ is complex character taking values in $\Q$. Such a character can be obtained by taking the sum of Galois conjugates of an irreducible complex character, and we will refer to these as irreducible rational characters even though they are not irreducible as complex characters. The irreducible rational characters form a $\Q$-basis for the $\Q$-vector space of rational characters.

Note that the representation corresponding to a rational character may not take values in $\GL_n(\Q)$ even though the character takes values in $\Q$ (the difference between the field of character values and the field in which matrix entries lie is measured by the Schur index).

The value taken by a rational character is constant on each rational conjugacy classes.