Question

I am always impressed by proofs that reach outside the obvious tool-kit. For example, the proof that the dimensions of the irreducible representations of a finite group divide the order of the group relies on the fact that the character values are algebraic integers. In particular, given a finite group $|G|$ and an irreducible character $\chi$ of dimension $n,$ $$\frac{1}{n} \sum_{s \in G} \chi(s^{-1})\chi(s) = \frac{|G|}{n}.$$ However, since $\frac{|G|}{n}$ is an algebraic integer (it is the image of an algebra homomorphism) lying in $\mathbb{Q},$ it in fact lies in $\mathbb{Z}.$