I read about semigroups.
Looks like sage could be useful for computing in algebraic number fields:
Read a very interesting page about Voevodsky, there was a mistake in one of his papers and it took 15 years to notice - he hopes to use computers to check proofs so that this doesn't happen again: <https://archive.is/OExMB>
Let $\mathfrak q$ be a prime ideal of $\mathcal O_K$, define $$\mathcal O_{\mathfrak q} = \{ x/y \in K | x,y \in \mathcal O_K, y \not\in \mathfrak q \}$$ this is a ring since $yy' \not\in \mathfrak q$. The set $\mathfrak q \mathcal O_{\mathfrak q}$ is the maximal ideal of this local ring. It's easily seen to be an ideal since $\mathfrak q$ is, to show it's maximal we show the quotient is a field, for uniqueness.. not sure yet.
Let $x/y$ be the representative for a nonzero element of $\mathcal O_{\mathfrak q}/\mathfrak q \mathcal O_{\mathfrak q}$, being nonzero implies $x \not\in \mathfrak q$ so we can just multiply by $y/x \in \mathcal O_{\mathfrak q}$ to get 1.
So how is it that an ideal becomes principal inside the localizaion? Let's do examples. Friend showed me this:
The simplest possible example of a non-principal ideal is (2,1+sqrt(-5)) in Z[sqrt(-5)]. 2/(1+sqrt(-5)) = <rationalize denominator> 2(1-sqrt(-5)) / 6 = (1-sqrt(-5))/3 3 is not in (2,1+sqrt(-5)) [why?], so (1-sqrt(-5))/3 is in O_q unfortunately (1+sqrt(-5))/2 isn't in O_q, the reason being that v_q(2) = 2, not 1 so 1+sqrt(-5) is a uniformizer, but 2 is not. you can check that q^2 = (2) the simplest unramified example is (3,1+sqrt(-5)).
For that one we find
? Mod(3/(1+x),x^2+5) %10 = Mod(-1/2*x + 1/2, x^2 + 5)
but dividing the other way doesn't seem to work, I expected it to.
You can prove unique factorization of prime ideals directly/globally and then use that to define the discrete valuation $v$. Or you can define the localization ot a place and work hard on that to show it's a DVR and get $v$, then unique factorization of ideals follows without much hard work.
It's very very hard though.
I'm reading bits of Atiya-MacDonald, Matsumura, Serre, various notes found online. Having a hard time settling on an approach and proof for this. It's more difficult than I want it to be no matter what way I choose.
Several detailled proofs here <http://math.stackexchange.com/questions/188585/localizations-of-dedekind-domains-are-discrete-valuation-rings>