P-adic numbers

For every prime number p, the p-adic numbers form an extension of the rational numbers first described by [Kurt Hensel]? in 1897. They have been used to solve several problems in number theory, many of them using [Helmut Hasse]?'s [local-global principle]?, which roughly states that an equation can be solved over the rational numbers if it can be solved over the real numbers and over the p-adic numbers for every prime p.

If p is a fixed prime number, then any integer can be written as a p-adic expansion in the form

        n
     ±  ∑  aipi
       i=0

where the a_i are integers in {0,...,p-1}. For example, the 2-adic or binary expansion of 35 is 1*2⁵ + 0*2⁴ + 0*2³ + 0*2² + 1*2¹ + 1*2⁰, often written in the shorthand notation 100011₂.

If we extend this set by allowing infinite sums of the form

        ∞
     ±  ∑  aipi
       i=0

we obtain the p-adic integers. Intuitively, these are numbers whose p-adic expansion to the left never stops. The main technical problem is to define a proper notion of infinite sum which makes these expressions meaningful; two different but equivalent solutions to this problem will be presented below.

The p-adic integers form a ring, and by taking quotients of two p-adic integers, we obtain the field Q_p of p-adic numbers. These numbers can be visualized as p-adic expansions which to the left never stop, and which have finitely many p-adic digits to the right of the "decimal" point. Q_p which has the nice topological property of completeness. This allows the development of [p-adic analysis]? akin to real analysis.

Construction

Analytic approach

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 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 uniquely specified by this requirement); now define |x|_p = p^-n. We also define |0|_p = 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. The p-adic norm defines a metric on Q by setting d_p(x,y) = |x - y|_p.

The field Q_p of p-adic numbers can then be defined as the completion of the metric space (Q,d_p); its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains Q.

Algebraic approach

In the algebraic approach, we first define the ring of p-adic integers, and then construct the field of quotients of this ring to get the field of p-adic numbers.

We start with the inverse limit of the rings Z_pⁿ (see modular arithmetic): a p-adic integer is then 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, see modular arithmetic.

Every natural number, m defines such a sequence (m mod pⁿ), and can therefore be regarded as a p-adic integer.

Note that every sequence (a_n) where the first element is not 0 has an inverse: since in that case, for every n, a_n and p are relatively prime (their greatest common divisor is a₁), and so a_n and pⁿ are relatively prime. Therefore, a_n has an inverse mod pⁿ, and the sequence of inverses, (b_n), is the sought inverse of (a_n).

The ring of p-adic integers has no zero divisors, so we can take the quotient field to get the field Q_p of p-adic numbers. Note that in the quotient field, every number can be written as p^-na with an element a in the ring and n a natural number.

Properties

The p-adic numbers contain the rational numbers Q and form a field of characteristic 0. This field cannot be turned into an ordered field.

The topology of the set of p-adic integers is that of a Cantor set; the topology of the set of p-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity).

As metric spaces, both the p-adic integers and the p-adic numbers are complete.

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 p-adic numbers that makes it algebraically closed has infinite degree; and for any given p, there exist infinitely many inequivalent algebraic extensions.