http://www.artofproblemsolving.com/blog/54443 defines them in the body of the first solution and gives some examples to how it works. Note that it helps on OMO #48 because if its an algebraic integer.

Apparently its taught well in Problems from the Book but unfortunately I don't own a copy of that book.

PFTB doesn't actually teach valuations explicitly, but SFTB does in the algebraic NT chapter (based on this, which solves one of the later problems in the original PFTB chapter). Actually I guess I'll write out the problem:
BTW, take the definition of with a grain of salt; I just copied it from U160 here and I have a feeling harazi might not have been too careful/precise writing it. I think in this specific case it's easier to rephrase it as " is an invertible algebraic integer for primitive th roots of unity " (to prove it, note that is an invertible algebraic integer for all relatively prime to ). I asked Aaron Pixton at MOP this year if that definition was correct and he said you need to look at the minimal polynomial in the field of -adic numbers rather than just for it to work. So here it's the same but in general probably not; also this was in a rather informal setting (the class was about Newton polygons as applied to irreducibility) so he might've left out some details.