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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 Qp of all p-adic numbers has the nice topological property of completeness, which allows the development of p-adic analysis[?] akin to real analysis.

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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>
where the ai 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:

<math>\pm\sum_{i=-\infty}^n a_i p^i</math>
A definite meaning is given to these sums based on Cauchy sequences using the p-adic 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 ai = 0 for all i<0.

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>
where k is some (not neccessarily positive) integer, we obtain the field Qp of p-adic numbers. Those p-adic numbers for which ai = 0 for all i<0 are also called the p-adic integers.

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.


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|_{\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 dp on Q by setting

The field Qp of p-adic numbers can then be defined as the completion of the metric space (Q,dp); 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 Qp, every element x may be written in a unique way as

<math>\sum_{i=k}^{\infty} a_i p^i</math>
where k is some integer and each ai is in {0,...,p-1}. This series converges to x with respect to the metric dp.

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 Zpn (see modular arithmetic): a p-adic integer is then a sequence (an)n≥1 such that an is in Zpn, and if n < m, an = am (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 (an) where the first element is not 0 has an inverse: since in that case, for every n, an and p are relatively prime (their greatest common divisor is a1), and so an and pn are relatively prime. Therefore, each an has an inverse mod pn, and the sequence of these inverses, (bn), is the sought inverse of (an).

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 Qp 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.


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, a quadratic extension is already algebraically closed. By contrast, the algebraic closure of the p-adic numbers has infinite degree. Furthermore, Qp 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 ep 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 Qp. For instance, the function f(x) = (1/|x|p)2 has zero derivative.

Given any elements r, r2, r3, r5, r7, ... where rp is in Qp (and Q stands for R), it is possible to find a sequence (xn) in Q such that for all p (including ∞), the limit of xn in Qp is rp.

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 ordP(x) for the exponent of P in this factorization, and define

<math>|x|_P = (NP)^{-\operatorname{ord}_P(x)}</math>
where NP denotes the (finite) cardinality of D/P. Completing with respect to this norm |.|P then yields a field EP, 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[?].

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