Odd-order groups have no nontrivial real irreducible characters
= Odd-order groups have no nontrivial real irreducible characters
For a real-valued <irreducible character> of a finite odd-order group, pairing $g$ with $g^{-1}$ gives $\langle\chi,1_G\rangle=(\chi(1)+2A)/|G|$, with $A$ an <algebraic integer>. If this inner product were zero, $\chi(1)$ would be even, since a rational <algebraic integer> is an integer. But an <irreducible character degree divides the group order>, so it is odd. Therefore the character is trivial.