5 - The p-adic numbers

Summary

The idea of p-adic numbers is due to Hensel, who was inspired by local power series expansions of meromorphic functions (see Warner 1989 p. 469f, Ebbinghaus et al. 1991 Chapter 6 and Ullrich 1998). We treat the p-adic numbers as relatives of the real numbers. In fact, completion of the rational field ℚ with respect to an absolute value leads either to the reals ℝ or to a field ℚ_p of p-adic numbers, where p is a prime number (see 44.9, 44.10, 51.4, 55.4), and these fields are locally compact. We consider the additive and the multiplicative group of ℚ_p in Sections 52 and 53, and we study squares and quadratic forms over ℚ_p in Section 54. It turns out (see 53.2) that the additive and the multiplicative group of ℚ_p are locally isomorphic, in the sense that some open (and compact) subgroup of ℚ^×_p is isomorphic to an open subgroup of ℚ⁺_p; this is similar to the situation for ℝ.

Comparing ℝ and ℚ_p, it appears that the structure of ℚ_p is dominated much more by algebraic and number theoretic features. A major topological difference between the locally compact fields ℝ and ℚ_p is the fact that ℝ is connected and ℚ_p is totally disconnected (51.10). Moreover, ℚ_p cannot be made into an ordered field (54.2).

In Sections 55–58 we put the fields ℚ_p in the context of general topological fields: we study absolute values, valuations and the corresponding topologies. Section 58 deals with the classification of all locally compact fields and skew fields.