P-adic numbers

The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers. However, the definition of a Cauchy sequence relies on the metric or norm chosen and, by choosing a different norm, numbers other than the real numbers can be constructed. The usual metric is called the Euclidean metric.

For a given prime p > 1, we define the p-adic metric in Q as follows.

For a non-zero rational number x, write x = pⁿy where neither the numerator and denominator of y have the factor p (notice that n is well defined); now define |x| = p^-n. We also define |0| = 0. It can be proved that all norms on Q are equivalent to either the Euclidean norm or one of the p-adic norms for some prime p.

A more intuitive description of p-adic numbers is the following. Integers can be written base p; elements of Z[1/p] (rational numbers whose denominator is a power of p) have expansions base p with a finite number of places to the right and left of the period. Real numbers are obtained by allowing for infinite expansions to the right; p-adic numbers are obtained by allowing for infinite expansions to the left.

Another way of defining the p-adic numbers is as inverse limits of Z_pⁿ: a p-adic number is a sequence a_n such that a_n is in Z_pⁿ, and if n<m, a_n = a_m (mod pⁿ). Note that addition and multiplication of such sequences is well defined, since addition and multiplication commute with the mod operator. Every natural number, m defines such a sequence {m mod pⁿ}. Note that every sequence where the first element is not 0 has an inverse: since in that case, for every n, a_n and p are mutually prime (their GCD is a₁, and so a_n and pⁿ are mutually prime. Therefore, a_n has an inverse mod p, and the sequence of inverses, {b_n}, is the inverse. This ring has no zero divisors, so we can take the quotient field. Note that in the quotient field, every number can be written as p^-na for a in the ring and n non-negative.

Once the p-adic numbers have been defined, for some p, then [p-adic analysis]? can be developed based on them. (Just as [real analysis]? can be developed once the real numbers have been defined.) The topology of the set of p-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity).

The real numbers have only a single algebraic extension, the complex numbers; in other words, a quadratic extension is already algebraically closed. The necessary algebraic extension to the field of the p-adic numbers that makes it algebraically closed has infinite degree; and for any given p, there exist infinitely many inequivalent algebraic extensions.