P-adic number

The p-adic number systems were first described by Kurt Hensel in 1897. For each prime number p, the p-adic number system extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The main use of these other systems is in number theory.

The extension is achieved by an alternative interpretation of the concept of absolute value. The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.

More formally, for a given prime p, the field Q_p of p-adic numbers is an extension field of the rational numbers. If all of the fields Q_p are collectively considered, we arrive at Helmut Hasse's local-global principle, which roughly states that certain equations can be solved over the rational numbers if and only if they can be solved over the real numbers and over the p-adic numbers for every prime p. The field Q_p is also given a topology derived from a metric, which is itself derived from an alternative valuation on the rational numbers. This metric is complete in the sense that every Cauchy sequence converges. This is what allows the development of calculus on Q_p, and it is the interaction of this analytic and algebraic structure which gives the p-adic number systems their power and utility.

In the context of elliptic curves p-adic numbers are usually referred to as $\ell$-adic numbers, due to the work of Jean-Pierre Serre. The prime p is often reserved for modular arithmetic of such curves.

Motivation

The simplest introduction to p-adic numbers is to consider 10-adic numbers, which are simply integers in which you allow an infinite number of digits to the left, for example, the number ...9999, and then do arithmetic with such numbers as usual. In other words, do arithmetic like you would with real numbers, but with digits going off to the left instead of to the right. The references to valuations and metrics given below are simply technical devices which justify the ordinary operations. For example, one has the computation

${\frac {...9999 \atop +1}{...000}}$

which is true because there are an infinite number of carries which never end, so there will never be a digit "1" on the left in the result. So a first 10-adic result is that ...999 = −1. It follows from this that negative integers can be represented as digit expansions in which all lefthand digits are eventually equal to 9. Computer scientists might be reminded of two's complement notation, in which negative integers are coded with the leftmost bit being set to 1: in the 2-adic integers, negative integers will correspond to numbers in which all lefthand digits are eventually equal to 1 (in general, p − 1 for p-adic numbers).

One point that confuses many people is why the p in p-adic numbers is always prime. As seen above, it is not absolutely necessary, as things work well enough in base 10. (Often the term g-adic number is used when the concept is used for a fixed composite number g. for example by Kurt Mahler). However, p-adic numbers are most useful for doing calculus-type computations, and it is important to always be able to divide, that is, one wants to be able to work in a field. The point is that p-adic numbers form a field if and only if p is a prime power, and you get the same result for a prime power as you do for the prime (e.g., base 16 is just shorthand for base 2). In particular, if p is not a prime power, then you can always find two nonzero p-adic numbers A and B such that AB = 0, which removes all possibility of finding their inverses. It is an interesting exercise to find such numbers for p = 10, for example, the following (check that the products are well defined over the 10-adics):

$A=\prod _{n=1}^{\infty }[2\,(2^{-1}{\rm {mod}}\,5^{n})],\qquad B=\prod _{n=1}^{\infty }[5\,(5^{-1}{\rm {mod}}\,2^{n})].$

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

$\pm \sum _{i=0}^{n}a_{i}p^{i}$

where the a_i are integers in {0,...,p − 1}. This is expressed by saying that the integer has been "written in base p". 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, to the reals) is to include sums of the form:

$\pm \sum _{i=-\infty }^{n}a_{i}p^{i}$

A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as 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

$\sum _{i=k}^{\infty }a_{i}p^{i}$

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. (Note: Z_p is often used to represent the set of integers modulo p. If each set is needed, the latter is usually written Z/pZ or Z/p. Be sure to check the notation for any text you read.)

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 ...1313132, i.e. the limit of the sequence 2, 32, 132, 3132, 13132, 313132, 1313132, ... Informally, we can see that multiplying this "infinite sum" by 3 in base 5 gives ...0000001. 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

$|x|_{2}=2$

$|x|_{3}=1/9$

$|x|_{5}=25$

$|x|_{7}=1/7$

$|x|_{11}=11$

$|x|_{\mbox{any other prime}}=1$

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

$\sum _{i=k}^{\infty }a_{i}p^{i}$

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, 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 coprime, 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 sequence.

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. Also contrasting the case of real numbers, the algebraic closure of Q_p is not (metrically) complete. Its (metric) completion is called Ω_p. Here an end is reached, as Ω_p is algebraically closed.

The field Ω_p is isomorphic to the field C of complex numbers, so we may regard Ω_p as the complex numbers endowed with an exotic metric. It should be noted that the existence of such a field isomorphism relies on the axiom of choice, and no explicit isomorphism can be given.

The p-adic numbers contain the nth cyclotomic field if and only if n divides p − 1. For instance, the nth cyclotomic field is a subfield of Q₁₃ iff n = 1, 2, 3, 4, 6, or 12.

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 the algebraic closure of p-adic numbers for all p.

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: Q_p → Q_p, f(x) = (1/|x|_p)² for x ≠ 0, f(0) = 0,

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

$|x|_{P}=(NP)^{-\operatorname {ord} _{P}(x)}$

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
• p-adic division algorithm