 Solution

The row orthogonality relations for a character table state that for irreducible characters $\chi ,\psi$,Equivalently, if $c$ runs through representatives of the conjugacy classes $C$,

Fix an irreducible $\chi$. The sum of the elements in a conjugacy class $C$ is central in $\mathbb{C}G$, so Schur lemma says that it acts in the representation affording $\chi$ by the scalarThis central character value of a conjugacy-class sum is an algebraic integer: the class sum acts by a matrix with integer entries on the regular representation, and ${\omega }_{\chi }(C)$ is one of its eigenvalues. Also $\overline{\chi (c)}$ is an algebraic integer because character values are sums of roots of unity.

Row orthogonality with $\psi =\chi$ now givesThe right-hand side is an algebraic integer. The left-hand side is rational, and every rational algebraic integer is an integer. Therefore