P-adic number 

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

<math>\pm\sum_{i=0}^n a_i p^i</math>

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:

<math>\pm\sum_{i=-\infty}^n a_i p^i</math>

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

<math>\sum_{i=k}^{\infty} a_i p^i</math>

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. 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.

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

<math>|x|_2=2</math>

<math>|x|_3=1/9</math>

<math>|x|_5=25</math>

<math>|x|_7=1/7</math>

<math>|x|_{11}=11</math>

<math>|x|_{\mbox{any other prime}}=1</math>

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

<math>d_p(x,y)=|x-y|_p</math>

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

<math>\sum_{i=k}^{\infty} a_i p^i</math>

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)_n≥1 such that a_n is in Z_pⁿ, 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 isn't 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, 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 isn't; 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, a 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, isn't 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 isn't true over Q_p. For instance, the function f(x) = (1/|x|_p)² has zero derivative.

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

<math>|x|_P = (NP)^{-\operatorname{ord}_P(x)}</math>

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[?].

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