Schur index of irreducible character is one in any prime characteristic

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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 Schur index (?) of \chi is 1.

Related facts

Facts used

  1. Every finite division ring is a field

Proof

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