TITLE: Why are the characters of the symmetric group integer-valued?
QUESTION [46 upvotes]: I remember one of my professors mentioning this fact during a class I took a while back, but when I searched my notes (and my textbook) I couldn't find any mention of it, let alone the proof.
My best guess is that it has something to do with Galois theory, since it's enough to prove that the characters are rational - maybe we have to find some way to have the symmetric group act on the Galois group of a representation or something. It would be nice if an idea along these lines worked, because then we could probably generalize to draw conclusions about the field generated by the characters of any group. Is this the case?

REPLY [2 votes]: Galois theory is probably the best and simplest explanation, and the most amenable to generalization for other groups (and more general number fields). This is already well-expounded in earlier answers. I just want to point out that it was already known to Frobenius that all irreducible characters of $S_{n}$ are $\mathbb{Z}$-combinations of permutation characters ( ie characters induced from trivial characters of subgroups). More precise information became available later (Young tableaux, etc, as mentioned in other answers), but what Frobenius knew already covered the question.