ABSTRACT
In classical algebraic geometry (i.e. over the field o f complex numbers) we can use transcendental methods. This means that we regard a rational function as an analytic function (o f one or more complex variables) and consider its power series expansion about a point. In abstract algebraic geometry the best we can do is to consider the corresponding formal power series. This is not so powerful as in the holomorphic case but it can be a very useful tool. The process of replacing polynomials by formal power series is,an example o f a general device known as completion. Another important instance of completion occurs in number theoiy in the formation of p-adic numbers. I f p is a prime number in Z we can work in the various quotient rings Z/pnZ : in other words, we can try and solve congruences modulo pn for higher and higher values o f n. This is analogous to the successive approximations given by the terms o f a Taylor expansion and, just as it is convenient to introduce formal power series, so it is convenient to introduce the p-adic numbers, these being the limit in a certain sense o f ZfpnZ asw->oo. In one respect, however, the p-adic numbers are more complicated than formal power series (in, say, one variable x). Whereas the polynomials of degree n are naturally embedded in the power series, the group ZjpnZ cannot be embedded in Z. Although a /»-adic integer can be thought of as a power series 2 anpn (0 < a » < p) this representation does not behave well under the ring operations.
