ABSTRACT
In classical algebraic geometry (i.e. over the field of complex numbers) we can use transcendental methods. This means that we regard a rational function as an analytic function (of 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 of a general device known as completion. Another important instance of completion occurs in number theory in the formation of p-adic numbers. If p is a prime number in Z we can work in the various quotient rings Z/p n Z: in other words, we can try and solve congruences modulo p n for higher and higher values of n. This is analogous to the successive approximations given by the terms of 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 of Z/p n Z as n → ∞. 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 Z/p n Z cannot be embedded in Z. Although a p-adic integer can be thought of as a power series ∑ a n p n ( 0 ≤ a n < p ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493621/dc273cdd-1d90-454a-a54f-66eaaaa4fe16/content/eq2248.tif"/> this representation does not behave well under the ring operations.
