Vector Spaces and Field Extensions

Abstract

Up to now, we have kept our attention focused on the field ℝ and its p-adic completions. We have already felt, however, the need to consider other fields (for example, when we dealt with the zeros of a function defined by a power series). In fact, just as we have emphasized the natural analogy between the p-adic fields ℚ_p and the field ℝ of real numbers, it is a very natural thing to do to look for an extension of ℚ_p that is analogous to the complex numbers. In other words, we would like to look for ways to extend ℚ_p in order to obtain a field that is not only complete (so that we can do analysis), but also algebraically closed (so that all polynomials have roots). This turns out to be more subtle (and therefore more interesting) than one might expect. It turns out, first of all, that to get an algebraically closed field one must make a very large extension of ℚ_p. This extension turns out not to be complete any more, so there is no other recourse but to go through the completion process again, and this finally yields the field we wanted. This is very different from the classical case, where going from ℝ to an algebraically closed field is just a small step (just add i), and the resulting field (the complex numbers) is already complete. The goal of this chapter is to tell thep-adic version of this story in its entirety.