p-adic Numbers

The set Qp 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 Qp is different from R (giving the first example of the dependency of the completion on the underlying metric), but Qp, 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 = ±pnp1n1p2n2 ·...· pmnm = pn (u / v),

where p and pk are distinct primes and the exponents n and nk 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 dp(u, v) = |u - v|p and, subsequently, to the completion Qp of Q. The members of Qp 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 Zp (not to be confused with the field of residues modulo p which uses the same notation.) The set of integers Z is dense in Zp, 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 Zp. 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,
 ...
 xn= 1 + p + p² + ... + pn,
 ...

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

 |xn - xm|p= |pm+1 + ... + pn|p
  = p-(m+1)

which tends to 0 as m (and hence n) grows. What p-adic number does {xk} 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

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Copyright © 1996-2017 Alexander Bogomolny

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 {xk} 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.

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