So far, I’ve covered unique factorization into ideals and the finiteness of the class group. As I noted a few eons ago, elementary algebraic number theory consists of three main results: unique factorization into ideals, the finiteness of and bound on the class group, and Dirichlet’s theorem on units.
Normally I’d just prove Dirichlet’s theorem, but I can’t quite tie it into elementary number theory the way I did ideal theory by considering the equation x^2 + 5 = y^3. Besides, the only proof of it that I’ve seen is too geometric, and is a nightmare by the standards of the previous proofs I’ve given.
So instead, I can go in one of several directions. The topic the textbooks talk about right after the above elementary concepts is valuations. In a nutshell, valuations are absolute values you can impose on a field, with the usual properties: |x||y| = |xy|, |x| >= 0 with equality iff x = 0, and |x| + |y| <= |x + y|.
The obvious valuation on Q is the normal absolute value, but there are others, in particular the p-adic valuations, which send a number of the form (p^n)a/b where p doesn’t divide a or b to 1/(p^n). In that case, the numbers that are close to 0 are high powers of p, rather than 1/n for large n.
Alternatively, in an email exchange, Katie mentioned to me that the things she’d read about the global field concept are very exciting. A global field is either a number field, or a function field. A function field is a finite extension of some F(p)(t). F(p) is simply Z/pZ regarded as a field; K(t) is the field of all quotients of polynomials with coefficients in K. Number fields and function fields exhibit many interesting analogies, especially around valuations.
At the same time, I could go in another direction and fill in the holes I’ve left in my posts. For example, I could prove that in every Dedekind domain, every ideal has at most two generators (but note it’s very different from saying the class number is at most 2), which all the textbooks I’ve looked at prove right after unique factorization into ideals. Or I could lose my sanity and ramble about norms and discriminants.
Yet another possible direction is to bite a bullet and develop the geometric concepts required for Dirichlet’s theorem. The operative term is Minkowski theory, or Minkowski’s theorem(s). The first major theorem is that in the vector space R^n, any convex body that’s symmetric around the origin and has volume greater than 2^n contains a point of Z^n other than the zero vector. That’s actually fairly simple to prove; the real annoyance comes from a theorem on matrices that can then be applied to the matrix of the conjugates of an integral basis of a number field K.
I could even abandon algebraic number theory entirely for the time being. There are related areas I could deal with, like Galois theory (whence comes my claim that if a function in K is invariant under conjugation, its values are in Q). Or I could go even further away and do things like my posts on the convoy problem and on the coin problem.
I know not that many people read my math posts, but still: it’s mostly up to you.