p-adic numbers

1 p-adic numbers

Let m be an integer, m > 1. An m-adic number is an expression a = ∑_{i=i_{0}}^{∞} a_{i}m^{i} where the digits a_{i} are in {0, 1, ..., m − 1}. Clearly, the m-adic numbers form a ring. An m-adic number a is called an m-adic integer if a_{i} = 0 for i < 0. Clearly, the m-adic integers form a ring.

Let p be prime. The p-adic numbers form a field called Q_{p}. The p-adic integers form a ring called Z_{p}. An element z ∈ Z_{p} is a unit (i.e., has an inverse in this ring) iff z_{0} ≠ 0.

Every nonzero p-adic number x can be written uniquely as p^{e}z where z is a unit in Z_{p}. Define the p-adic norm | |_{p} by |x|_{p} = p^{−e} if x = p^{e}z ≠ 0 and |0|_{p} = 0. Let | |_{∞} be the ordinary absolute value. Then for any nonzero rational number r we have ∏_{v} |r|_{v} = 1 if v runs over ∞ and all primes p. Just like R is the completion of Q for the norm | |_{∞}, the fields Q_{p} are the completions of Q for the norms | |_{p}.

(The completion of a metric space is obtained by adding new points that are the limits of Cauchy sequences.)

For the metric d(x, y) = |x−y|_{p}, the partial sums of a = ∑_{i=i_{0}}^{∞} a_{i}m^{i} converge to a.

Example Let m = 10. Write a 10-adic integer as a (possibly left-infinite) sequence of decimal digits. We have ...99999 = −1, for example because ...99999 + 1 = 0 (or because, summing a geometric progression, ...99999 = 9/(1 − 10) = −1).

Example Let m = 10. We can find two nonzero 10-adic integers ...32 and ...25 with product 0 by making sure that one is divisible by arbitrarily high powers of 2 and the other by arbitrarily high powers of 5. This is achieved by growing longer and longer tails, each time prefixing a digit with the property that there are infinitely many powers of 2 (or 5) that end with the tail thus obtained.

Example Compute 1/10 in the 5-adic numbers: \frac{1}{10} = ...2222.3. Compute 1/7 in the 5-adic integers: \frac{1}{7} = ...2412032412033. Clearly, the rational numbers are precisely the p-adic numbers that are eventually periodic.

In order to solve equations over the p-adic numbers, one first solves them mod p, and then lifts the obtained solution(s) to solutions mod p^{i} for all i.

For example, x^{2} = 2 has no solutions in Q_{5} since it has no solutions mod 5. (So, Q_{5} is not algebraically closed.) But x^{2} = −1 has two solutions in Q_{5}. First of all, 2^{2} = 3^{2} = −1 (mod 5). Next, if x^{2} + 1 = 0 (mod 5^{i}) where i > 0, say x^{2} +1 = a.5^{i}, put y = x+c.5^{i}. Then y^{2} +1 = x^{2} +1+2cx.5^{i} = (a+2cx).5^{i} (mod 5^{i+1}), and we can pick c (in a unique way) so as to make this 0 (mod 5^{i+1}). This argument works more generally, and the lemma describing it is called Hensel’s Lemma.

Both Z_{p} and Q_{p} have the cardinality of the continuum, that is, of R. The ring Z_{p} (with product topology) is compact. The field Q_{p} is locally compact.

The various extensions Q_{p} of Q are independent in the sense that given numbers x_{p} ∈ Q_{p} and x_{∞} ∈ R there is a sequence (r_{i})_{i} of rational numbers such that for each v this sequence has limit x_{v} in Q_{v} (with Q_{∞} = R).