Last time we examined Dedekind domains in finite separable field extensions. One advantage to using a separable field extension that we did not use is that we can base extend to a finite Galois extension, where as we see the action of Galois forces the splitting of primes to be very uniform. (more…)
July 14, 2009
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July 8, 2009
Last time we took a look at Dedekind domains with fraction fields and found that if was any finite field extension of that the integral closure of in is Dedekind. The proof in this case is somewhat involved, but becomes slightly less so and shows that is also a finitely generated -module under the assumption that is also separable. In the following post we look at some of what makes life so much easier in the separable case. (more…)
July 2, 2009
As has been hinted in many previous posts, many facts about algebraic number theory tell us about geometric objects like elliptic curves. For instance, if you are working on a problem which primarily uses the affine geometry of a curve like the semistable reduction theorem for elliptic curves, the scheme you’re working on is opposite to what’s called a Dedekind Domain. We begin a series of posts on Dedekind Domains, beginning today with the very abstract and progressing to the concrete(which would of course be terrible for teaching this material but I mean these posts as more of a reference work).