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

q = ±pⁿp₁^n₁p₂^n₂ ·...· p_m^n_m = pⁿ (u / v),

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,

|1001/1000 - 1|₂ = |1/1000|₂ = |2^-35^-3|₂ = 8.

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:

	1.	\|u\|_p = 0 only if u = 0.
	2.	\|uv\|_p = \|u\|_p\|v\|_p, for any rational u and v
	3.	\|u + v\|_p ≤ \|u\|_p + \|v\|_p, for any rational u and v

In fact the triangle inequality in 3. can be strengthened

3'.

|u + v|_p ≤ max{|u|_p, |v|_p}, for any rational u and v.

An even stronger result holds:

3''.

For |u|_p ≠ |v|_p, |u + v|_p = max{|u|_p, |v|_p}, for any rational u and v.

To prove the latter, assume |u|_p < |v|_p. Then, by 3', |u + v|_p ≤ |v|_p. On the other hand, |v|_p = |u + v - u|_p ≤ max{|u + v|_p, |u|_p} = |u + v|_p, due to our assumption.

Norm |.|_p is called the p-adic norm (or p-adic absolute value). It was introduced by Kurt Hensel, a student of Leopold Kronecker, in 1902.

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

	x₀	= 1,
	x₁	= 1 + p,
	x₂	= 1 + p + p²,
	x₃	= 1 + p + p² + p³,
	...
	x_n	= 1 + p + p² + ... + pⁿ,
	...

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

	\|x_n - x_m\|_p	= \|p^m+1 + ... + pⁿ\|_p
		= p^-(m+1)

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

E. B. Berger, Exploring the Number Jungle, AMS, 2000
J. R. Goldman, The Queen of Mathematics, A K Peters, 1998
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

= 1 + p + ... + p^m + ...

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 + ... + p^m + ...
	pq	= p + p² + ... + p^m+1 + ...
	pq + 1	= 1 + p + ... + p^m + ... = 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.

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