p-adic number

In mathematics, and chiefly number theory, the p-adic 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 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 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 p-adic number systems their power and utility.

The p in p-adic is a variable and may be replaced with a constant (yielding, for instance, "the 2-adic numbers") or another placeholder variable (for expressions such as "the ℓ-adic numbers").

Introduction

This section is an informal introduction to p-adic numbers, using examples from the ring of 10-adic numbers. (Base 10 was chosen to highlight the analogy with decimals. The 10-adic numbers are generally not used in mathematics: since 10 is not prime, the 10-adics 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 non-terminating decimal as follows

$\frac{1}{3}=0.333333\ldots.$

Informally, non-terminating 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.

10-adic numbers use a similar non-terminating 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 10-adic expansions are close if they differ by a large positive power of 10. Thus 3333 and 4333, which differ by 10³, are close in the 10-adic metric, and 33333333 and 43333333 are even closer, differing by 10⁷.

In the 10-adic metric, the following sequence of numbers gets closer and closer to −1:

$9=-1+10 \,$

$99=-1+10^2 \,$

$999=-1+10^3 \,$

$9999=-1+10^4 \,$

and taking this sequence to its limit, we can say (informally) that the 10-adic expansion of −1 is

$\dots 9999=-1.\,$

In this notation, 10-adic 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 p-adic numbers—for alternatives see the Notation section below.

More formally, a 10-adic number can be defined as

$\sum_{i=n}^\infty a_i 10^i$

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 10-adic expansions that are identical to their decimal expansions. Other numbers may have non-terminating 10-adic expansions.

It is possible to define addition, subtraction, and multiplication on 10-adic numbers in a consistent way, so that the 10-adic numbers form a commutative ring.

We can create 10-adic expansions for negative numbers as follows

$-100 = -1 \times 100 = \dots 9999 \times 100 = \dots 9900 \,$

$\Rightarrow -35 = -100+65 = \dots 9900 + 65 = \dots 9965 \,$

$\Rightarrow -\left(3+\dfrac{1}{2}\right)=\dfrac{-35}{10}= \dfrac{\dots 9965}{10}=\dots 9996.5$

and fractions which have non-terminating decimal expansions also have non-terminating 10-adic expansions. For example

$\dfrac{10^6-1}{7}=142857; \dfrac{10^{12}-1}{7}=142857142857; \dfrac{10^{18}-1}{7}=142857142857142857$

$\Rightarrow-\dfrac{1}{7}=\dots 142857142857142857$

$\Rightarrow-\dfrac{6}{7}=\dots 142857142857142857 \times 6 = \dots 857142857142857142$

$\Rightarrow\dfrac{1}{7} = -\dfrac{6}{7}+1 = \dots 857142857142857143.$

Generalizing the last example, we can find a 10-adic expansion for any rational number p⁄q such that q is co-prime to 10; Euler's theorem guarantees that if q is co-prime to 10, then there is an n such that 10ⁿ − 1 is a multiple of q.

However, 10-adic numbers have one major drawback. It is possible to find pairs of non-zero 10-adic numbers whose product is 0. In other words, the 10-adic 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.

p-adic expansions

If p is a fixed prime number, then any positive integer can be written in a base p expansion in the form

$\sum_{i=0}^n a_i p^i$

where the a_i are integers in {0, …, p − 1}. For example, the 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 extending this description to the larger domain of the rationals (and, ultimately, to the reals) is to use 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 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

$\sum_{i=k}^{\infty} a_i p^i$

where k is some (not necessarily positive) integer, we obtain the p-adic expansions defining 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. (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 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,… . 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.

While it is possible to use this approach to rigorously define p-adic 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 p-adic metric. Two different but equivalent solutions to this problem are presented in the Constructions section below.

Notation

There are several different conventions for writing p-adic expansions. So far this article has used a notation for p-adic expansions in which powers of p increase from right to left. With this right-to-left notation the 3-adic expansion of ¹/₅, for example, is written as

$\dfrac{1}{5}=\dots 121012102_3.$

When performing arithmetic in this notation, digits are carried to the left. It is also possible to write p-adic expansions so that the powers of p increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of ¹/₅ is

$\dfrac{1}{5}=2.01210121\dots_3\mbox{ or }\dfrac{1}{15}=20.1210121\dots_3.$

p-adic expansions may be written with other sets of digits instead of {0, 1, …, p − 1}. For example, the 3-adic expansion of ¹/₅ can be written using balanced ternary digits {1,0,1} as

$\dfrac{1}{5}=\dots\underline{1}11\underline{11}11\underline{11}11\underline{1}_3.$

In fact any set of p integers which are in distinct residue classes modulo p may be used as p-adic digits. In number theory, Teichmüller digits are sometimes used.

Constructions

Analytic approach

See also: p-adic order

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… . 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 p-adic absolute value 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 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 3² 5⁻² 7 11⁻¹

$\displaystyle|x|_2=2 \,\!$

$\displaystyle|x|_3=1/9 \,\!$

$|x|_5=25 \,\!$

$\displaystyle|x|_7=1/7 \,\!$

$|x|_{11}=11 \,\!$

$|x|_{\text{any other prime}}=1. \,\!$

This definition of |x|_p has the effect that high powers of p become "small". By the fundamental theorem of arithmetic, for a given non-zero rational number x there is a unique finite set of distinct primes $p_1, \ldots, p_r$ and a corresponding sequence of non-zero integers $a_1, \ldots, a_r$ such that:

$|x| = p_1^{a_1}\ldots p_r^{a_r}.$

It then follows that $|x|_{p_i} = p_i^{-a_i}$ for all $1\leq i\leq r$, and $|x|_p = 1\,$ for any other prime $p \notin \{p_1,\ldots p_r\}.$

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 p-adic absolute values for some prime p. The p-adic absolute value 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.

With this absolute value, the field Q_p is a local field.

Algebraic approach

In the algebraic approach, we first define the ring of p-adic integers, and then construct the field of fractions 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, then a_n ≡ a_m (mod pⁿ).

Every natural number m defines such a sequence (a_n) by a_n = 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, …).

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 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). For example, consider the p-adic integer corresponding to the natural number 7; as a 2-adic number, it would be written (1, 3, 7, 7, 7, 7, 7, ...). This object's inverse would be written as an ever-increasing sequence that begins (1, 3, 7, 7, 23, 55, 55, 183, 439, 439, 1463 ...). Naturally, this 2-adic 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 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 field of fractions to get the field Q_p of p-adic numbers. Note that in this field of fractions, every non-integer p-adic number can be uniquely written as p⁻ⁿu with a natural number n and a unit in the p-adic integers u. This means that

$\mathbf{Q}_p=\left(\mathbf{Z}_p\right)^{\mbox{frac}}\cong (p^{\mathbf{Z}})^{-1}\mathbf{Z}_p.$

Note that S ^{− 1}A, where $S=p^{\mathbf{Z}}=\{p^{n}:n\in\mathbf{Z}\}$ 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 p-adic integers is the inverse limit of the finite rings Z/p^kZ, 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 p-group of rank n, denoted Z(p^∞)ⁿ, is the ring of n×n matrices over the p-adic integers; this is sometimes referred to as the Tate module.

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.

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 p-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity).^[5] 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.^[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 p-adic 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 p-adic 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 Q₁₃ if and only if n = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicative p-torsion in the p-adic numbers, if p > 2. Also, -1 is the only non-trivial torsion element in 2-adic numbers.

Given a natural number k, the index of the multiplicative group of the k-th powers of the non-zero 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 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.^[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 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.^[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_∞, 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.

The field Q_p is a locally compact Hausdorff space.

If $\mathbf{K}$ is a finite Galois extension of $\mathbf{Q}_{p}$, the Galois group $\text{Gal}(\mathbf{K}/\mathbf{Q}_{p})$ is solvable. Thus, the Galois group $\text{Gal}(\overline{\mathbf{Q}}_{p}/\mathbf{Q}_{p})$ is prosolvable.

Rational arithmetic

Hehner and Horspool proposed in 1979 the use of a p-adic 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 2ⁿ−1 is a Mersenne prime, its reciprocal will require 2ⁿ−1 bits to represent.

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 field of fractions. Pick a non-zero prime ideal P of D. 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 non-zero 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

$|x|_P = c^{-\operatorname{ord}_P(x)}.$

Completing with respect to this absolute value |.|_P yields a field E_P, the proper generalization of the field of p-adic 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 non-trivial non-Archimedean absolute value on E arises as some |.|_P. The remaining non-trivial absolute values on E arise from the different embeddings of E into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of E into the fields C_p, thus putting the description of all the non-trivial 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.

Local-global principle

Helmut Hasse's local-global 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 p-adic numbers for every prime p.

See also

• Hensel's lemma
• Mahler's theorem
• C-minimal theory

Notes

1. ^ Hensel, Kurt (1897). "Über eine neue Begründung der Theorie der algebraischen Zahlen". Jahresbericht der Deutschen Mathematiker-Vereinigung 6 (3): 83–88. http://www.digizeitschriften.de/resolveppn/GDZPPN00211612X&L=2. 
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.
3. ^ Finding explicit examples of zero divisors is surprisingly difficult. See Gérard Michon's article at [1]
4. ^ Robert (2000) Section 1.1
5. ^ Robert (2000) Section 2.3
6. ^ Gouvêa (2000) Corollary 3.3.8
7. ^ Gouvêa (2000) Corollary 5.3.10
8. ^ Gouvêa (2000) Theorem 5.7.4
9. ^ Gouvêa (2000) Proposition 5.7.8
10. ^ Gouvêa (2000) Proposition 3.4.2
11. ^ Robert (2000) Section 4.1
12. ^ Robert (2000) Section 5.1
13. ^ Eric C. R. Hehner, R. Nigel Horspool, A new representation of the rational numbers for fast easy arithmetic. SIAM Journal on Computing 8, 124-134. 1979.

References

• Gouvêa, Fernando Q. (2000). p-adic Numbers : An Introduction (2nd ed.). Springer. ISBN 3540629114. 
• Koblitz, Neal (1996). P-adic Numbers, p-adic Analysis, and Zeta-Functions (2nd ed.). Springer. ISBN 0387960171. 
• Robert, Alain M. (2000). A Course in p-adic Analysis. Springer. ISBN 0387986693. 
• Bachman, George (1964). Introduction to p-adic Numbers and Valuation Theory. Academic Press. ISBN 0120702681. 
• Steen, Lynn Arthur (1978). Counterexamples in Topology. Dover. ISBN 048668735X. 

External links

• Weisstein, Eric W., "p-adic Number" from MathWorld.
• p-adic integers on PlanetMath
• p-adic number at Springer On-line Encyclopaedia of Mathematics
• Completion of Algebraic Closure - on-line lecture notes by Brian Conrad
• An Introduction to p-adic Numbers and p-adic Analysis - on-line lecture notes by Andrew Baker, 2007