P-adic number

For every prime number p, the p-adic numbers form an extension field 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 and only if it can be solved over the real numbers and over the p-adic numbers for every prime p. The space Q_p of all p-adic numbers has the nice topological property of completeness, which allows the development of p-adic analysis akin to real analysis.

Motivation

If p is a fixed prime number, then any integer can be written as a p-adic expansion (usually referred to as writing the number in "base p") in the form

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.

The familiar approach to generalizing this description to the larger domain of the rationals (and, ultimately, the real numbers) is to include sums of the form:

A definite meaning is given to these sums based on Cauchy sequences using the familiar Euclidean metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...₅. In this formulation, the integers are precisely those numbers which can be represented in the form where a_i = 0 for all i<0.

As an alternative, if we extend the p-adic expansions by allowing infinite sums of the form

where k is some (not necessarily positive) integer, we obtain the field Q_p of p-adic numbers. Those p-adic numbers for which a_i = 0 for all i<0 are also called the p-adic integers. The p-adic integers form a subring of Q_p, denoted Z_p.

Intuitively, as opposed to p-adic expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p (as is done for the real numbers as described above), these are numbers whose p-adic expansion to the left are allowed to go on forever. For example, the p-adic expansion of 1/3 in base 5 is the limit of the sequence ...31313132₅. Informally, we can see that multiplying this "infinite sum" by 3 in base 5 gives ...0000001₅. As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 is a p-adic integer in base 5.

The main technical problem is to define a proper notion of infinite sum which makes these expressions meaningful - this requires the introduction of the p-adic metric. Two different but equivalent solutions to this problem are presented below.

Constructions

Analytic approach

The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; this allows us to, for example, write 1 as 1.000... = 0.999... . However, the definition of a Cauchy sequence relies on the metric chosen and, by choosing a different one, numbers other than the real numbers can be constructed. The usual metric which yields the real numbers is called the Euclidean metric.

For a given prime p, we define the p-adic metric in Q as follows: for any non-zero rational number x, there is a unique integer n allowing us to write x = pⁿ(a/b), where neither of the integers a and b is divisible by p. Unless the numerator or denominator of x contains a factor of p, n will be 0. Now define |x|_p = p^-n. We also define |0|_p = 0.

For example with x = 63/550 = 2^-1 3² 5^-2 7 11^-1

| x | ₂ = 2

| x | ₃ = 1 / 9

| x | ₅ = 25

| x | ₇ = 1 / 7

| x | ₁₁ = 11

| x | _{any other prime} = 1

This definition of |x|_p has the effect that high powers of p become "small".

It can be proved that each norm on Q is equivalent either to the Euclidean norm or to one of the p-adic norms for some prime p. The p-adic norm defines a metric d_p 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.

It can be shown that in Q_p, every element x may be written in a unique way as

where k is some integer and each a_i is in {0,...,p-1}. This series converges to x with respect to the metric d_p.

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/pⁿZ (see modular arithmetic): a p-adic integer is then a sequence (a_n)_n≥1 such that a_n is in Z/pⁿZ, and if n < m, a_n = a_m (mod pⁿ).

Every natural number m defines such a sequence (m mod pⁿ), and can therefore be regarded as a p-adic integer. For example, in this case 35 as a 2-adic integer would be written as the sequence {1, 3, 3, 3, 35, 35, 35, ...}.

Note that pointwise addition and multiplication of such sequences is well defined, since addition and multiplication commute with the mod operator, see modular arithmetic. Also, 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, and so a_n and pⁿ are relatively prime. Therefore, each a_n has an inverse mod pⁿ, and the sequence of these inverses, (b_n), is the sought inverse of (a_n).

Every such sequence can alternatively be written as a series of the form we considered above. For instance, in the 3-adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2*3 + 0*3² + 1*3³ + 0*3⁴ + ... The partial sums of this latter series are the elements of the given series.

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 this quotient field, every number can be uniquely written as p⁻ⁿu with a natural number n and a p-adic integer u.

Properties

The set of p-adic integers is uncountable.

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). In particular, the space of p-adic integers is compact while the space of p-adic numbers is not; it is only locally compact. As metric spaces, both the p-adic integers and the p-adic numbers are complete.

The real numbers have only a single proper algebraic extension, the complex numbers; in other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of the p-adic numbers has infinite degree. Furthermore, Q_p has infinitely many inequivalent algebraic extensions.

The number e, defined as the sum of reciprocals of factorials, is not a member of any p-adic field; but e^p is a p-adic number for all p except 2, for which one must take at least the fourth power. Thus e is a member of all algebraic extensions of p-adic numbers.

Over the reals, the only functions whose derivative is zero are the constant functions. This is not true over Q_p. For instance, the function f(x) = (|x|_p)² has zero derivative everywhere but is not even locally constant at 0.

Given any elements r_∞, r₂, r₃, r₅, r₇, ... where r_p is in Q_p (and Q_∞ stands for R), it is possible to find a sequence (x_n) in Q such that for all p (including ∞), the limit of x_n in Q_p is r_p.

Generalizations and related concepts

The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way. This will be described now.

Suppose D is a Dedekind domain and E is its quotient field. The non-zero prime ideals of D are also called finite places or finite primes of E. If x is a non-zero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of finite primes of E. If P is such a finite prime, we write ord_P(x) for the exponent of P in this factorization, and define

where NP denotes the (finite) cardinality of D/P. Completing with respect to this norm |.|_P then yields a field E_P, the proper generalization of the field of p-adic numbers to this setting.

Often, one needs to simultaneously keep track of all the above mentioned completions, which are seen as encoding "local" information. This is accomplished by adele rings and idele groups.

See also

Mahler's theorem