 padic number

In mathematics, and chiefly number theory, the padic number system for any prime number p 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 extension is achieved by an alternative interpretation of the concept of absolute value.
First described by Kurt Hensel in 1897,^{[1]} the padic 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 padic analysis essentially provides an alternative form of calculus.
More formally, for a given prime p, the field Q_{p} of padic numbers is a completion of the rational numbers. 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 space is complete in the sense that every Cauchy sequence converges to a point in Q_{p}. 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.
The p in padic is a variable and may be replaced with a constant (yielding, for instance, "the 2adic numbers") or another placeholder variable (for expressions such as "the ℓadic numbers").
Contents
Introduction
This section is an informal introduction to padic numbers, using examples from the ring of 10adic numbers. (Base 10 was chosen to highlight the analogy with decimals. The 10adic numbers are generally not used in mathematics: since 10 is not prime, the 10adics are not a field.) More formal constructions and properties are given below.
In the standard decimal representation, almost all^{[2]} real numbers do not have a terminating decimal representation. For example, 1/3 is represented as a nonterminating decimal as follows
Informally, nonterminating decimals are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. If two decimal expansions differ only after the 10th decimal place, they are quite close to one another; and if they differ only after the 20th decimal place, they are even closer.
10adic numbers use a similar nonterminating expansion, but with a different concept of "closeness" called a metric. Whereas two decimal expansions are close to one another if they differ by a large negative power of 10, two 10adic expansions are close if they differ by a large positive power of 10. Thus 3333 and 4333, which differ by 10^{3}, are close in the 10adic metric, and 33333333 and 43333333 are even closer, differing by 10^{7}.
In the 10adic metric, the following sequence of numbers gets closer and closer to −1:
and taking this sequence to its limit, we can say (informally) that the 10adic expansion of −1 is
In this notation, 10adic expansions can be extended indefinitely to the left, in contrast to decimal expansions, which can be extended indefinitely to the right. Note that this is not the only way to write padic numbers—for alternatives see the Notation section below.
More formally, a 10adic number can be defined as
where each of the a_{i} is a digit taken from the set {0, 1, …..., 9} and the initial index n may be positive, negative or 0, but must be finite. From this definition, it is clear that positive integers and positive rational numbers with terminating decimal expansions will have terminating 10adic expansions that are identical to their decimal expansions. Other numbers may have nonterminating 10adic expansions.
It is possible to define addition, subtraction, and multiplication on 10adic numbers in a consistent way, so that the 10adic numbers form a commutative ring.
We can create 10adic expansions for negative numbers as follows
and fractions which have nonterminating decimal expansions also have nonterminating 10adic expansions. For example
Generalizing the last example, we can find a 10adic expansion for any rational number p⁄q such that q is coprime to 10; Euler's theorem guarantees that if q is coprime to 10, then there is an n such that 10^{n} − 1 is a multiple of q.
However, 10adic numbers have one major drawback. It is possible to find pairs of nonzero 10adic numbers whose product is 0. In other words, the 10adic numbers are not a domain because they contain zero divisors.^{[3]} This turns out to be because 10 is a composite number which is not a power of a prime. This problem is avoided by using a prime number p as the base of the number system instead of 10.
padic expansions
If p is a fixed prime number, then any positive integer can be written in a base p expansion in the form
where the a_{i} are integers in {0, …, p − 1}. For example, the 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 extending this description to the larger domain of the rationals (and, ultimately, to the reals) is to use sums of the form:
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 for which a_{i} = 0 for all i < 0.
As an alternative, if we extend the base p expansions by allowing infinite sums of the form
where k is some (not necessarily positive) integer, we obtain the padic expansions defining 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}. (Not to be confused with the ring of integers modulo p which is also sometimes written Z_{p}. To avoid ambiguity, Z/pZ or Z/(p) are often used to represent the integers modulo p.)
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,… . 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.
While it is possible to use this approach to rigorously define padic numbers and explore their properties, just as in the case of real numbers other approaches are generally preferred. Hence we want to define a notion of infinite sum which makes these expressions meaningful, and this is most easily accomplished by the introduction of the padic metric. Two different but equivalent solutions to this problem are presented in the Constructions section below.
Notation
There are several different conventions for writing padic expansions. So far this article has used a notation for padic expansions in which powers of p increase from right to left. With this righttoleft notation the 3adic expansion of ^{1}/_{5}, for example, is written as
When performing arithmetic in this notation, digits are carried to the left. It is also possible to write padic expansions so that the powers of p increase from left to right, and digits are carried to the right. With this lefttoright notation the 3adic expansion of ^{1}/_{5} is
padic expansions may be written with other sets of digits instead of {0, 1, …, p − 1}. For example, the 3adic expansion of ^{1}/_{5} can be written using balanced ternary digits {1,0,1} as
In fact any set of p integers which are in distinct residue classes modulo p may be used as padic digits. In number theory, Teichmüller digits are sometimes used.
Constructions
Analytic approach
See also: padic orderThe 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… . The definition of a Cauchy sequence relies on the metric chosen, though, so if we choose a different one, we can construct numbers other than the real numbers. The usual metric which yields the real numbers is called the Euclidean metric.
For a given prime p, we define the padic absolute value 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 in lowest terms contains p as a factor, 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}
This definition of x_{p} has the effect that high powers of p become "small". By the fundamental theorem of arithmetic, for a given nonzero rational number x there is a unique finite set of distinct primes and a corresponding sequence of nonzero integers such that:
It then follows that for all , and for any other prime
It is a theorem of Ostrowski that each absolute value on Q is equivalent either to the Euclidean absolute value, the trivial absolute value, or to one of the padic absolute values for some prime p. The padic absolute value defines a metric d_{p} on Q by setting
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
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}.
With this absolute value, the field Q_{p} is a local field.
Algebraic approach
In the algebraic approach, we first define the ring of padic integers, and then construct the field of fractions 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, then a_{n} ≡ a_{m} (mod p^{n}).
Every natural number m defines such a sequence (a_{n}) by a_{n} = 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, …).
The operators of the ring amount to pointwise addition and multiplication of such sequences. This is well defined because addition and multiplication commute with the mod operator, see modular arithmetic.
Moreover, every sequence (a_{n}) where the first element is not 0 has an inverse. 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}). For example, consider the padic integer corresponding to the natural number 7; as a 2adic number, it would be written (1, 3, 7, 7, 7, 7, 7, ...). This object's inverse would be written as an everincreasing sequence that begins (1, 3, 7, 7, 23, 55, 55, 183, 439, 439, 1463 ...). Naturally, this 2adic integer has no corresponding natural number.
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 sequence.
The ring of padic integers has no zero divisors, so we can take the field of fractions to get the field Q_{p} of padic numbers. Note that in this field of fractions, every noninteger padic number can be uniquely written as p^{−n}u with a natural number n and a unit in the padic integers u. This means that
Note that S ^{− 1}A, where is a multiplicative subset (contains the unit and closed under multiplication) of a commutative ring with unit A, is an algebraic construction called the ring of fractions of A by S.
Properties
The ring of padic integers is the inverse limit of the finite rings Z/p^{k}Z, but is nonetheless uncountable,^{[4]} and has the cardinality of the continuum. Accordingly, the field Q_{p} is uncountable. The endomorphism ring of the Prüfer pgroup of rank n, denoted Z(p^{∞})^{n}, is the ring of n×n matrices over the padic integers; this is sometimes referred to as the Tate module.
The padic numbers contain the rational numbers Q and form a field of characteristic 0. This field cannot be turned into an ordered field.
Let the topology τ on Z_{p} be defined by taking as a basis all sets of the form U_{a}(n) = {n + λ p^{a} for λ in Z_{p} and a in N}. Then Z_{p} is a compactification of Z, under the derived topology (it is not a compactification of Z with its usual topology). The relative topology on Z as a subset of Z_{p} is called the 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).^{[5]} 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.^{[6]}
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.^{[7]} 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.^{[8]} Its (metric) completion is called C_{p}. Here an end is reached, as C_{p} is algebraically closed.^{[9]}
The field C_{p} is isomorphic to the field C of complex numbers, so we may regard C_{p} as the complex numbers endowed with an exotic metric. It should be noted that the proof of existence of such a field isomorphism relies on the axiom of choice, and does not provide an explicit example of such an isomorphism.
The padic numbers contain the nth cyclotomic field (n>2) if and only if n divides p − 1.^{[10]} For instance, the nth cyclotomic field is a subfield of Q_{13} if and only if n = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicative ptorsion in the padic numbers, if p > 2. Also, 1 is the only nontrivial torsion element in 2adic numbers.
Given a natural number k, the index of the multiplicative group of the kth powers of the nonzero elements of Q_{p} in the multiplicative group of Q_{p} is finite.
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.^{[11]} (Thus a number with similar properties as e  namely a pth root of e^{p}  is a member of the algebraic closure of the 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}.^{[12]} 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}.
The field Q_{p} is a locally compact Hausdorff space.
If is a finite Galois extension of , the Galois group is solvable. Thus, the Galois group is prosolvable.
Rational arithmetic
Hehner and Horspool proposed in 1979 the use of a padic representation for rational numbers on computers.^{[13]} The primary advantage of such a representation is that addition, subtraction, and multiplication can be done in a straightforward manner analogous to similar methods for binary integers; and division is even simpler, resembling multiplication. However, it has the disadvantage that representations can be much larger than simply storing the numerator and denominator in binary; for example, if 2^{n}−1 is a Mersenne prime, its reciprocal will require 2^{n}−1 bits to represent.
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 field of fractions. Pick a nonzero prime ideal P of D. 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 nonzero prime ideals of D. We write ord_{P}(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set
Completing with respect to this absolute value ._{P} yields a field E_{P}, the proper generalization of the field of padic numbers to this setting. The choice of c does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when the residue field D/P is finite, to take for c the size of D/P.
For example, when E is a number field, Ostrowski's theorem says that every nontrivial nonArchimedean absolute value on E arises as some ._{P}. The remaining nontrivial absolute values on E arise from the different embeddings of E into the real or complex numbers. (In fact, the nonArchimedean absolute values can be considered as simply the different embeddings of E into the fields C_{p}, thus putting the description of all the nontrivial absolute values of a number field on a common footing.)
Often, one needs to simultaneously keep track of all the above mentioned completions when E is a number field (or more generally a global field), which are seen as encoding "local" information. This is accomplished by adele rings and idele groups.
Localglobal principle
Helmut Hasse's localglobal principle is said to hold for an equation if it can be solved over the rational numbers if and only if it can be solved over the real numbers and over the padic numbers for every prime p.
See also
Notes
 ^ Hensel, Kurt (1897). "Über eine neue Begründung der Theorie der algebraischen Zahlen". Jahresbericht der Deutschen MathematikerVereinigung 6 (3): 83–88. http://www.digizeitschriften.de/resolveppn/GDZPPN00211612X&L=2.
 ^ The number of real numbers with terminating decimal representations is countably infinite, while the number of real numbers without such a representation is uncountably infinite.
 ^ Finding explicit examples of zero divisors is surprisingly difficult. See Gérard Michon's article at [1]
 ^ Robert (2000) Section 1.1
 ^ Robert (2000) Section 2.3
 ^ Gouvêa (2000) Corollary 3.3.8
 ^ Gouvêa (2000) Corollary 5.3.10
 ^ Gouvêa (2000) Theorem 5.7.4
 ^ Gouvêa (2000) Proposition 5.7.8
 ^ Gouvêa (2000) Proposition 3.4.2
 ^ Robert (2000) Section 4.1
 ^ Robert (2000) Section 5.1
 ^ Eric C. R. Hehner, R. Nigel Horspool, A new representation of the rational numbers for fast easy arithmetic. SIAM Journal on Computing 8, 124134. 1979.
References
 Gouvêa, Fernando Q. (2000). padic Numbers : An Introduction (2nd ed.). Springer. ISBN 3540629114.
 Koblitz, Neal (1996). Padic Numbers, padic Analysis, and ZetaFunctions (2nd ed.). Springer. ISBN 0387960171.
 Robert, Alain M. (2000). A Course in padic Analysis. Springer. ISBN 0387986693.
 Bachman, George (1964). Introduction to padic Numbers and Valuation Theory. Academic Press. ISBN 0120702681.
 Steen, Lynn Arthur (1978). Counterexamples in Topology. Dover. ISBN 048668735X.
External links
 Weisstein, Eric W., "padic Number" from MathWorld.
 padic integers on PlanetMath
 padic number at Springer Online Encyclopaedia of Mathematics
 Completion of Algebraic Closure  online lecture notes by Brian Conrad
 An Introduction to padic Numbers and padic Analysis  online lecture notes by Andrew Baker, 2007
Number systems Countable sets Real numbers and
their extensionsOther number systems  Cardinal numbers
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 padic numbers
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