P-adic number 1

P-adic number

The 3-adic integers, with selected correspondingcharacters on their Pontryagin dual group

In mathematics the p-adic number system for any prime number pextends the ordinary arithmetic of the rational numbers in a waydifferent from the extension of the rational number system to the realand complex number systems. The extension is achieved by analternative interpretation of the concept of "closeness" or absolutevalue. In particular, p-adic numbers have the interesting property thatthey are said to be close when their difference is divisible by a highpower of p – the higher the power the closer they are. This propertyenables p-adic numbers to encode congruence information in a waythat turns out to have powerful applications in number theoryincluding, for example, in the famous proof of Fermat's Last Theoremby Andrew Wiles.[1]

p-adic numbers were first described by Kurt Hensel in 1897,[2] thoughwith hindsight Wikipedia:Avoid weasel words some of Kummer'searlier work can be interpreted as implicitly using p-adic numbers. Wikipedia:Avoid weasel words The p-adicnumbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods intonumber theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentiallyprovides an alternative form of calculus.

More formally, for a given prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The fieldQp is also given a topology derived from a metric, which is itself derived from an alternative valuation on therational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp.This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraicstructure 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") oranother placeholder variable (for expressions such as "the ℓ-adic numbers").

p-adic expansionsWhen dealing with natural numbers, if we take p to be a fixed prime number, then any positive integer can bewritten as a base p expansion in the form

where the ai 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:

A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as metric. Thus, forexample, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...5. In this formulation, theintegers are precisely those numbers for which ai = 0 for all i < 0.With p-adic numbers, on the other hand, we choose to extend the base p expansions in a different way. Because in the p-adic world high positive powers of p are small and high negative powers are large, we consider infinite sums

P-adic number 2

of the form:

where k is some (not necessarily positive) integer. With this approach we obtain the p-adic expansions of the p-adicnumbers. Those p-adic numbers for which ai = 0 for all i < 0 are also called the p-adic integers.As opposed to real number expansions which extend to the right as sums of ever smaller, increasingly negativepowers of the base p, p-adic numbers may expand to the left forever, a property that can often be true for the p-adicintegers. For example, consider the p-adic expansion of 1/3 in base 5. It can be shown to be …13131325, i.e., thelimit of the sequence 25, 325, 1325, 31325, 131325, 3131325, 13131325, … :

Multiplying this infinite sum by 3 in base 5 gives …00000015. As there are no negative powers of 5 in thisexpansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 satisfies the definition of being ap-adic integer in base 5.More formally, the p-adic expansions can be used to define the field Qp of p-adic numbers while the p-adicintegers form a subring of Qp, denoted Zp. (Not to be confused with the ring of integers modulo p which is alsosometimes written Zp. To avoid ambiguity, Z/pZ or Z/(p) are often used to represent the integers modulo p.)While it is possible to use the approach above to define p-adic numbers and explore their properties, just as in thecase of real numbers other approaches are generally preferred. Hence we want to define a notion of infinite sumwhich makes these expressions meaningful, and this is most easily accomplished by the introduction of the p-adicmetric. Two different but equivalent solutions to this problem are presented in the Constructions section below.

NotationThere are several different conventions for writing p-adic expansions. So far this article has used a notation forp-adic expansions in which powers of p increase from right to left. With this right-to-left notation the 3-adicexpansion 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 p-adicexpansions so that the powers of p increase from left to right, and digits are carried to the right. With thisleft-to-right notation the 3-adic expansion of 1⁄5 is

p-adic expansions may be written with other sets of digits instead of {0, 1, …, p − 1}. For example, the 3-adicexpansion 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 p-adic digits. In numbertheory, Teichmüller digits are sometimes used.

P-adic number 3

Constructions

Analytic approach

p = 2 ← distance = 1 →

De-ci-

mal

Bi-nary

← d = ½ → ← d = ½ →

‹ d=¼ › ‹ d=¼ › ‹ d=¼ › ‹ d=¼ ›

‹⅛› ‹⅛› ‹⅛› ‹⅛› ‹⅛› ‹⅛› ‹⅛› ‹⅛›

................................................

17 10001 J

16 10000 J

15 1111 L

14 1110 L

13 1101 L

12 1100 L

11 1011 L

10 1010 L

9 1001 L

8 1000 L

7 111 L

6 110 L

5 101 L

4 100 L

3 11 L

2 10 L

1 1 L

0 0…000 L

−1 1…111 J

−2 1…110 J

−3 1…101 J

−4 1…100 J

Dec Bin ················································

2-adic ( p = 2 ) arrangement of integers, from left to right. This shows a hierarchical subdivision patterncommon for ultrametric spaces. Points within a distance 1/8 are grouped in one colored strip. A pair of stripswithin a distance 1/4 has the same chroma, four strips within a distance 1/2 have the same hue. The hue isdetermined by the least significant bit, the saturation – by the next (21) bit, and the brightness depends on thevalue of 22 bit. Bits (digit places) which are less significant for the usual metric are more significant for the p-adicdistance.

P-adic number 4

Similar picture for p = 3 (click to enlarge) shows 3 closed balls ofradius 1/3, where each consists of 3 balls of 1/9

The real numbers can be defined as equivalenceclasses of Cauchy sequences of rational numbers; thisallows us to, for example, write 1 as 1.000… =0.999… . The definition of a Cauchy sequence relieson the metric chosen, though, so if we choose adifferent one, we can construct numbers other than thereal numbers. The usual metric which yields the realnumbers is called the Euclidean metric.

For a given prime p, we define the p-adic absolutevalue in Q as follows: for any non-zero rationalnumber x, there is a unique integer n allowing us towrite x = pn(a/b), where neither of the integers a and bis divisible by p. Unless the numerator ordenominator of x in lowest terms contains p as afactor, n will be 0. Now define |x|p = p−n. We alsodefine |0|p = 0.

For example with x = 63/550 = 2−1·32·5−2·7·11−1

This definition of |x|p has the effect that high powers of p become "small". By the fundamental theorem ofarithmetic, for a given non-zero rational number x there is a unique finite set of distinct primes and acorresponding sequence of non-zero 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, thetrivial absolute value, or to one of the p-adic absolute values for some prime p. So the only norms on Q moduloequivalence are the absolute value, the trivial absolute value and the p-adic absolute value which means that thereare only as many completions (with respect to a norm) of Q.The p-adic absolute value 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 areequivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges tozero. 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

P-adic number 5

where k is some integer such that ak ≠ 0 and each ai is in {0, …, p − 1 }. This series converges to x with respect tothe metric dp.With this absolute value, the field Qp is a local field.

Algebraic approachIn the algebraic approach, we first define the ring of p-adic integers, and then construct the field of fractions of thisring 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(an)n≥1 such that an is in Z/pnZ, and if n ≤ m, then an ≡ am (mod pn).Every natural number m defines such a sequence (an) by an = m mod pn and can therefore be regarded as a p-adicinteger. 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 definedbecause addition and multiplication commute with the "mod" operator, see modular arithmetic.Moreover, every sequence (an) where the first element is not 0 has an inverse. In that case, for every n, an and p arecoprime, and so an and pn are relatively prime. Therefore, each an has an inverse mod pn, and the sequence of theseinverses, (bn), is the sought inverse of (an). For example, consider the p-adic integer corresponding to the naturalnumber 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 anever-increasing sequence that begins (1, 3, 7, 7, 23, 55, 55, 183, 439, 439, 1463 ...). Naturally, this 2-adic integer hasno corresponding natural number.Every such sequence can alternatively be written as a series. 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 thegiven sequence.The ring of p-adic integers has no zero divisors, so we can take the field of fractions to get the field Qp of p-adicnumbers. Note that in this field of fractions, every non-integer p-adic number can be uniquely written as p−n u with anatural number n and a unit in the p-adic integers u. This means that

Note that S−1 A, where is a multiplicative subset (contains the unit and closed undermultiplication) of a commutative ring with unit , is an algebraic construction called the ring of fractions of by .

PropertiesThe ring of p-adic integers is the inverse limit of the finite rings Z/pkZ, but is nonetheless uncountable,[3] and has thecardinality of the continuum. Accordingly, the field Qp is uncountable. The endomorphism ring of the Prüferp-group of rank n, denoted Z(p∞)n, is the ring of n×n matrices over the p-adic integers; this is sometimes referred toas the Tate module.The p-adic numbers contain the rational numbers Q and form a field of characteristic 0. This field cannot be turnedinto an ordered field.Let the topology τ on Zp be defined by taking as a basis all sets of the form { n + λ pa for λ in Zp and a in N}. ThenZp is a compactification of Z, under the derived topology (it is not a compactification of Z with its usual discretetopology). The relative topology on Z as a subset of Zp is called the p-adic topology on Z.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).[4] 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

P-adic number 6

integers and the p-adic numbers are complete.[5]

The real numbers have only a single proper algebraic extension, the complex numbers; in other words, this quadraticextension is already algebraically closed. By contrast, the algebraic closure of the p-adic numbers has infinitedegree,[6] i.e. Qp has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers,although there is a unique extension of the p-adic valuation to the algebraic closure of Qp, it is not (metrically)complete.[7][8] Its (metric) completion is called Cp or Ωp.[8][9] Here an end is reached, as Cp is algebraicallyclosed.[8][10] Unlike the complex field, Cp is not locally compact.[9]

The field Cp is algebraically isomorphic to the field C of complex numbers, so we may regard Cp as the complexnumbers endowed with an exotic metric. It should be noted that the proof of existence of such a field isomorphismrelies 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.[11] For instance, the nthcyclotomic field is a subfield of Q13 if and only if n = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicativep-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 kth powers of the non-zero elements of Qp inthe multiplicative group of Qp is finite.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-adicnumber for all p except 2, for which one must take at least the fourth power.[12] (Thus a number with similarproperties as e – namely a pth root of ep – is a member of the algebraic closure of the p-adic numbers for all p.)For reals, the only functions whose derivative is zero are the constant functions. This is not true over Qp.[13] Forinstance, the function

f: Qp → Qp, 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 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.The field Qp is a locally compact Hausdorff space.

If is a finite Galois extension of , the Galois group is solvable. Thus, the Galois groupis prosolvable.

Rational arithmeticEric Hehner and Nigel Horspool proposed in 1979 the use of a p-adic representation for rational numbers oncomputers[14] called Quote notation. 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; anddivision is even simpler, resembling multiplication. However, it has the disadvantage that representations can bemuch larger than simply storing the numerator and denominator in binary; for example, if 2n − 1 is a Mersenneprime, its reciprocal will require 2n − 1 bits to represent.

Generalizations and related conceptsThe reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, forinstance 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-zeroelement of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powersof non-zero prime ideals of D. We write ordP(x) for the exponent of P in this factorization, and for any choice ofnumber c greater than 1 we can set

P-adic number 7

Completing with respect to this absolute value |.|P yields a field EP, the proper generalization of the field of p-adicnumbers to this setting. The choice of c does not change the completion (different choices yield the same concept ofCauchy sequence, so the same completion). It is convenient, when the residue field D/P is finite, to take for c the sizeof D/P.For example, when E is a number field, Ostrowski's theorem says that every non-trivial non-Archimedean absolutevalue on E arises as some |.|P. The remaining non-trivial absolute values on E arise from the different embeddings ofE into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply thedifferent embeddings of E into the fields Cp, thus putting the description of all the non-trivial absolute values of anumber field on a common footing.)Often, one needs to simultaneously keep track of all the above mentioned completions when E is a number field (ormore generally a global field), which are seen as encoding "local" information. This is accomplished by adele ringsand idele groups.

Local–global principleHelmut Hasse's local–global principle is said to hold for an equation if it can be solved over the rational numbers ifand only if it can be solved over the real numbers and over the p-adic numbers for every prime p. This principleholds e.g. for equations given by quadratic forms, but fails for higher polynomials in several indeterminates.

Notes[1] F. Q. Gouvêa, A Marvelous Proof, The American Mathematical Monthly, Vol. 101, No. 3 (Mar., 1994), pp. 203–222[3][3] Robert (2000) Section 1.1[4][4] Robert (2000) Section 2.3[5][5] Gouvêa (2000) Corollary 3.3.8[6][6] Gouvêa (2000) Corollary 5.3.10[7][7] Gouvêa (2000) Theorem 5.7.4[8][8] Cassels (1986) p.149[9][9] Koblitz (1980) p.13[10][10] Gouvêa (2000) Proposition 5.7.8[11][11] Gouvêa (2000) Proposition 3.4.2[12][12] Robert (2000) Section 4.1[13][13] Robert (2000) Section 5.1[14] 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• Bachman, George (1964). Introduction to p-adic Numbers and Valuation Theory. Academic Press.

ISBN 0-12-070268-1.• Cassels, J.W.S. (1986). Local Fields. London Mathematical Society Student Texts 3. Cambridge University Press.

ISBN 0-521-31525-5. Zbl  0595.12006 (http:/ / www. zentralblatt-math. org/ zmath/ en/ search/?format=complete& q=an:0595. 12006).

• Gouvêa, Fernando Q. (2000). p-adic Numbers : An Introduction (2nd ed.). Springer. ISBN 3-540-62911-4.• Koblitz, Neal (1980). p-adic analysis: a short course on recent work. London Mathematical Society Lecture Note

Series 46. Cambridge University Press. ISBN 0-521-28060-5. Zbl  0439.12011 (http:/ / www. zentralblatt-math.org/ zmath/ en/ search/ ?format=complete& q=an:0439. 12011).

• Koblitz, Neal (1996). P-adic Numbers, p-adic Analysis, and Zeta-Functions (2nd ed.). Springer.ISBN 0-387-96017-1.

• Robert, Alain M. (2000). A Course in p-adic Analysis. Springer. ISBN 0-387-98669-3.

P-adic number 8

• Steen, Lynn Arthur (1978). Counterexamples in Topology. Dover. ISBN 0-486-68735-X.

External links• Weisstein, Eric W., " p-adic Number (http:/ / mathworld. wolfram. com/ p-adicNumber. html)" from MathWorld.• p-adic integers (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=3118), PlanetMath.org.• p-adic number (http:/ / www. encyclopediaofmath. org/ index. php/ P-adic_number) at Springer On-line

Encyclopaedia of Mathematics• Completion of Algebraic Closure (http:/ / math. stanford. edu/ ~conrad/ 248APage/ handouts/ algclosurecomp.

pdf) – on-line lecture notes by Brian Conrad• An Introduction to p-adic Numbers and p-adic Analysis (http:/ / www. maths. gla. ac. uk/ ~ajb/ dvi-ps/

padicnotes. pdf) - on-line lecture notes by Andrew Baker, 2007• Efficient p-adic arithmetic (http:/ / homes. esat. kuleuven. be/ ~fvercaut/ talks/ pAdic. pdf) (slides)

Article Sources and Contributors 9

Article Sources and ContributorsP-adic number  Source: http://en.wikipedia.org/w/index.php?oldid=553107165  Contributors: 130.182.125.xxx, A5, Adam majewski, Amahoney, Anonymous Dissident, Arthur Rubin, AugPi,AxelBoldt, Ben Standeven, Bender235, Bgwhite, Bluap, Brentt, Bryan Derksen, CRGreathouse, Charles Matthews, Chas zzz brown, Chinju, Chowbok, Chris the speller, Ciphergoth,Classicalecon, Codygunton, Conversion script, Coopercc, Crasshopper, CryptoDerk, DFRussia, Damian Yerrick, Daqu, David Eppstein, Dcoetzee, DeaconJohnFairfax, Deltahedron,Dharma6662000, Dmcallas, Dnas, Dominus, Dratman, Drusus 0, Dysprosia, Długosz, E.V.Krishnamurthy, Eequor, ElNuevoEinstein, Emurphy42, Eric Drexler, Eric Kvaalen, Fropuff, Gandalf61,Gauge, Gene Ward Smith, Giftlite, Graham87, H00kwurm, Hairchrm, Hans Adler, Haziel, Heptadecagon, Ideyal, Ilanpi, Incnis Mrsi, Iseeaboar, Isnow, JackSchmidt, Jafet, Jallotta, Jason Quinn,Jbolden1517, JeffBurdges, Joriki, Jowa fan, Julian Brown, KSmrq, Keith Edkins, Khazar, Kier07, Kusma, Lambiam, Lethe, Linas, Loadmaster, LokiClock, Looxix, MFH, MarSch, Marozols,MathMartin, Mav, Maxal, Melchoir, Michael Hardy, Miguel, Mikolt, Minesweeper, MinorProphet, Mon4, Nbarth, Nippashish, Numericana, Oleg Alexandrov, Oli Filth, Patrick, Paul August,Paul Clapham, PaulTanenbaum, PierreAbbat, Pjacobi, Populus, Qpt, Quondum, R.e.b., ReiVaX, Revolver, Rill2503456, Roentgenium111, Rotem Dan, RxS, Singingwolfboy, SirJective, Sligocki,Stephen Bain, Tachyon², Taejo, TakuyaMurata, Thatcher, The Anome, TheBlueWizard, Tobias Bergemann, Toby Bartels, Tosha, Trovatore, Ttwo, Wadems, Waltpohl, Zundark, 151 anonymousedits

Image Sources, Licenses and ContributorsImage:3-adic integers with dual colorings.svg  Source: http://en.wikipedia.org/w/index.php?title=File:3-adic_integers_with_dual_colorings.svg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: User:MelchoirImage:3-adic metric on integers.svg  Source: http://en.wikipedia.org/w/index.php?title=File:3-adic_metric_on_integers.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:Incnis Mrsi

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/