5.1, due on October 10

I found this section to be easier to understand than the previous several sections for the most part. That being said, corollary 5.5, it's proof, and the following examples for it were rather confusing. I think I understand what the corollary is trying to say because I know the rule for the number of congruence classes in Z, but I don't understand how they word the description of the number and distinctness of the congruence classes. I don't understand how the congruence classes would be identified given a modulo because I don't feel like it's evident given the examples what all the possible remainders could be from dividing polynomials.

I'm curious to know the applicability of congruence classes of polynomials. It's interesting that congruence works the same in the ring of polynomials because it's a field. Then does that mean all fields have congruence modulo something? If so, are any of them important to look at? If they aren't important, what makes modulo integers and modulo polynomials more useful? These might be stupid questions but I feel like all of the discussion about congruence is something so interested but something I have never thought about or studied before and I feel like there must be a lot of applications or at least a few really important applications.