Complete fields and valuation rings

In order to make further progress in our investigation of finite extensions L/K of the fraction field K of a Dedekind domain A, and in particular, to determine the primes p of K that ramify in L, we introduce a new tool that allows us to “localize” fields. We have seen how useful it can be to localize the ring A at a prime ideal p: this yields a discrete valuation ring Ap, a principal ideal domain with exactly one nonzero prime ideal, which is much easier to study than A. By Proposition 2.6, the localizations of A at its prime ideals p collectively determine the ring A. Localizing A does not change its fraction field K. But there is an operation we can perform on K that is analogous to localizing A: we can construct the completion of K with respect to one of its absolute values. When K is a global field, this yields a local field (which we will define in the next lecture). At first glance taking completions might seem to make things more complicated, but as with localization, it actually simplifies matters by allowing us to focus on a single prime. For those who have not seen this construction before, we briefly review some background material on completions, topological rings, and inverse limits.