Schur index of irreducible character is one in any prime characteristic

Statement
Suppose $$p$$ is a prime number, $$G$$ is a finite group, and $$\chi$$ is the character of an an irreducible linear representation of $$G$$ in characteristic $$p$$. Then, the fact about::Schur index of $$\chi$$ is 1.

Related facts

 * Schur index of irreducible character in characteristic zero divides exponent
 * Schur index divides degree of irreducible representation
 * Odd-order p-group implies every irreducible representation has Schur index one

Facts used

 * 1) uses::Every finite division ring is a field

Proof
The proof follows from the alternative characterization of Schur index in terms of division rings.