an 2 00 3 A brief introduction to p - adic numbers Stephen Semmes

In this short survey we look at a few basic features of p-adic numbers, somewhat with the point of view of a classical analyst. In particular, with p-adic numbers one has arithmetic operations and a norm, just as for real or complex numbers. Let Z denote the integers, Q denote the rational numbers, R denote the real numbers, and C denote the complex numbers. Also let | · | denote the usual absolute value function or modulus on the complex numbers. On the rational numbers there are other absolute value functions that one can consider. Namely, if p is a prime number, define the p-adic absolute value function | · |p on Q by |x|p = 0 when x = 0, |x|p = p −k when x = pm/n, where k is an integer and m, n are nonzero integers which are not divisible by p. One can check that |xy|p = |x|p |y|p (1) and |x+ y|p ≤ |x|p + |y|p (2) for all x, y ∈ Q, just as for the usual absolute value function. In fact, |x+ y|p ≤ max(|x|p, |y|p) (3) for all x, y ∈ Q. This is called the ultrametric version of the triangle inequality. Just as the usual absolute value function leads to the distance function |x−y|, the p-adic absolute value function leads to the p-adic distance function |x − y|p on Q. With respect to this distance function, the rationals are not complete as a metric space, and one can complete the rationals to get a larger space Qp. This is analogous to obtaining the real numbers by completing the rationals with respect to the standard absolute value function. By standard