Odd-order p-group implies every irreducible representation has Schur index one

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This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
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Statement

Suppose p is an odd prime and G is a finite p-group, i.e., a group of prime power order where the underlying prime is p. Then, every irreducible representation of G over a splitting field has Schur index (?) 1, i.e., every irreducible representation can be realized over the field generated by the character values of the representation.

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