So I have a question about field extensions. This theorem is proving that if we have a polynomial field F[x] with an irreducible polynomial p(x) in it, then there is always another field that contains an isomorphic copy of F in which p(x) has a root. They essentially show that the quotient K=F[x]/(p(x)) works as the extension of F in which p(x) has a root. But I'm confused-- isn't a quotient field typically smaller than the field it came from? Because the thing it's "divided by" reduces everything by that, right? If it's smaller, how can it be a field extension? I know I'm missing something stupid, but I'd love it if someone could help me.
Another question: They introduce the concept of the minimal polynomial in F[x] for an element "a" (a member of the extension field of F, K) which is algebraic over F[x], saying that it is a unique monic irreducible polynomial in F[x] which has a as a root. My question is: how do you necessarily find this polynomial? I mean, for sqrt(2) over Q, it's pretty clear that it's x^2-2, but I don't see how you'd know for a more complicated element. For example, one question from the book asks to find the minimal polynomial for (1+i) over Q. I think I found that it is x^4 + 4 (because (i+1) plugged into that is a root, and that polynomial is definitely irreducible in Q), but I don't really know how I got that. I just kind of squared (1+i), then noticed if I squared it again I got -4.
Ahhh, I swear I had more questions that I've forgotten... Anyway, thanks in advance! I really need some things cleared up and my professor isn't very helpful.