Padic number
From Academic Kids

 The title of this article is incorrect because of technical limitations. The correct title is padic number. (With a lowercase and preferably italicized p.)
The padic number systems were first described by Kurt Hensel in 1897. For each prime p, the padic 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. This is achieved by an alternative interpretation of the concept of absolute value. The padic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory, but their influence now extends far beyond this. For example, the field of padic analysis essentially provides an alternative form of calculus.
More precisely, for a given prime p, the field Q_{p} of padic numbers is an extension of the rational numbers. If all of the fields Q_{p} are collectively considered, we arrive at Helmut Hasse's localglobal 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 padic 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 padic number systems their power and utility.
In the context of elliptic curves padic numbers are usually referred to as <math>\ell<math>adic numbers, due to the work of JeanPierre Serre.
Contents 
Motivation
The simplest introduction to padic numbers is to consider 10adic 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
 <math>
\frac{{...9999 \atop +1}} { ...000}
<math>
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 10adic 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 2adic integers, negative integers will correspond to numbers in which all lefthand digits are eventually equal to 1 (in general, p − 1 for padic numbers).
One point that confuses many people is why the p in padic numbers is always prime. As seen above, it is not absolutely necessary, as things work well enough in base 10. (Often the term gadic number is used when the concept is used for a fixed composite number g. for example by Kurt Mahler). However, padic numbers are most useful for doing calculustype 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 padic 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 padic 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 10adics):
 <math>
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)]. <math>
If p is a fixed prime number, then any integer can be written as a padic expansion (writing the number in "base p") in the form
 <math>\pm\sum_{i=0}^n a_i p^i<math>
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 2adic or binary expansion of 35 is 1·2^{5} + 0·2^{4} + 0·2^{3} + 0·2^{2} + 1·2^{1} + 1·2^{0}, often written in the shorthand notation 100011_{2}.
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:
 <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 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 padic expansions by allowing infinite sums of the form
 <math>\sum_{i=k}^{\infty} a_i p^i<math>
where k is some (not necessarily positive) integer, we obtain the field Q_{p} of padic numbers. Those padic numbers for which a_{i} = 0 for all i < 0 are also called the padic integers. The padic 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 padic 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 padic expansion to the left are allowed to go on forever. For example, the padic 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 padic 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 padic 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 padic metric in Q as follows: for any nonzero rational number x, there is a unique integer n allowing us to write x = p^{n}(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} 3^{2} 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 padic norms for some prime p. The padic norm defines a metric d_{p} on Q by setting
 <math>d_p(x,y)=xy_p<math>
The field Q_{p} of padic 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
 <math>\sum_{i=k}^{\infty} a_i p^i<math>
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 padic integers, and then construct the field of quotients of this ring to get the field of padic numbers.
We start with the inverse limit of the rings Z/p^{n}Z (see modular arithmetic): a padic integer is then a sequence (a_{n})_{n≥1} such that a_{n} is in Z/p^{n}Z, and if n < m, a_{n} = a_{m} (mod p^{n}).
Every natural number m defines such a sequence (m mod p^{n}), and can therefore be regarded as a padic integer. For example, in this case 35 as a 2adic 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 p^{n} are relatively prime. Therefore, each a_{n} has an inverse mod p^{n}, 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 3adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2*3 + 0*3^{2} + 1*3^{3} + 0*3^{4} + ... The partial sums of this latter series are the elements of the given series.
The ring of padic integers has no zero divisors, so we can take the quotient field to get the field Q_{p} of padic numbers. Note that in this quotient field, every number can be uniquely written as p^{−n}u with a natural number n and a padic integer u.
Properties
The set of padic integers is uncountable.
The padic 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 padic integers is that of a Cantor set; the topology of the set of padic numbers is that of a Cantor set minus a point (which would naturally be called infinity). In particular, the space of padic integers is compact while the space of padic numbers is not; it is only locally compact. As metric spaces, both the padic integers and the padic 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 padic 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 padic numbers contain the nth cyclotomic field if and only if n divides p − 1. For instance, the nth cyclotomic field is a subfield of Q_{13} 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 padic field; but e^{p} is a padic 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 padic 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_{∞}, r_{2}, r_{3}, r_{5}, r_{7}, ... 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 padic 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 nonzero prime ideals of D are also called finite places or finite primes of E. If x is a nonzero 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>
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 padic 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
Topics in mathematics related to quantity  
Numbers  Natural numbers  Integers  Rational numbers  Constructible numbers  Algebraic numbers  Computable numbers  Real numbers  Complex numbers  Splitcomplex numbers  Bicomplex numbers  Hypercomplex numbers  Quaternions  Octonions  Sedenions  Superreal numbers  Hyperreal numbers  Surreal numbers  Nominal numbers  Ordinal numbers  Cardinal numbers  padic numbers  Integer sequences  Mathematical constants  Large numbers  Infinity 
es:Número pádico fr:Nombre padique ja:P進数 ru:Pадическое число tr:Psel Sayılar