p-adic Numbers

The set Q_p of p-adic numbers is the completion of the space of all rationals Q when the latter is endowed with a special metric |.|_p. Not only Q_p is different from R (giving the first example of the dependency of the completion on the underlying metric), but Q_p, being defined for every prime p, supplies a sequence of distinct completions of the same set of the rationals.

First, for every prime p, we define norm |.|_p, on the set Q of rational numbers. Every nonzero rational number q = r/s admits a unique factorization in the form

where p and p_k are distinct primes and the exponents n and n_k may be positive or negative, depending on the prevalence of the power of the corresponsing prime in the numerator or denominator of q; u and v are integers prime to p.

Definition

|q|_p = p^-n,

where n is the exponent of p in the above factorization. Separately we also define |0|_p = 0. As an example,

Which shows that 1001/1000 and 1 are not at all close in Q₂. In Q₅, the situation is even worse: |1001/1000 - 1|₅ = 125. Note, however that for any integer n and prime p, |n|_p ≤ 1.

|.|_p has three important properties that qualify it as a norm:

In fact the triangle inequality in 3. can be strengthened

An even stronger result holds:

The availability of the triangle inequality leads to a definition of a metric d_p(u, v) = |u - v|_p and, subsequently, to the completion Q_p of Q. The members of Q_p are known as p-adic numbers. p-adic numbers u that satisfy |u|_p ≤ 1 are called p-adic integers. p-adic integers form a commutative ring Z_p (not to be confused with the field of residues modulo p which uses the same notation.) The set of integers Z is dense in Z_p, meaning that every p-adic integer is a limit (in the p-norm of an integer sequence.) Any (reduced) fraction whose denominator contains no factors of p belongs to Z_p. p is the only prime p-adic integer.

It is easy to give an example of an integer sequence which is Cauchy in the sense of the p-adic norm. The motivation comes from |pⁿ|_p = p^-n. Set

For the sequence {x_k}, assuming n > m,

which tends to 0 as m (and hence n) grows. What p-adic number does {x_k} converge to? This is a particular case of what is known as a p-adic expansion.

References

1. E. B. Berger, Exploring the Number Jungle, AMS, 2000
2. J. R. Goldman, The Queen of Mathematics, A K Peters, 1998
3. F. Q. Gouvêa, Local and Global in Number Theory, in The Princeton Companion to Mathematics T. Gowers (ed.), Princeton University Press, 2008

Strange as it may appear, we'll write

q

= 1 + p + ... + pm + ...

As we just said, q is well defined as a limit of the sequence {x_k} and this is how the identity should be understood. All arithmetic operations are permitted to carry over the limits so that

	q	= 1 + p + ... + pm + ...
	pq	= p + p² + ... + pm+1 + ...
	pq + 1	= 1 + p + ... + pm + ... = q

So that pq + 1 = q, implying q = 1 / (1 - p). Taking p = 2 we obtain:

-1

= 1 + 2 + 4 + ...

One certainly needs time and effort to get used to such weirdness.