Innumber theory, given aprime numberp,[note 1] thep-adic numbers form an extension of therational numbers that is distinct from thereal numbers, though with some similar properties;p-adic numbers can be written in a form similar to (possiblyinfinite)decimals, but with digits based on a prime numberp rather than ten, and extending to the left rather than to the right.
For example, comparing the expansion of the rational number inbase3 vs. the3-adic expansion,
Formally, given a prime numberp, ap-adic number can be defined as aserieswherek is aninteger (possibly negative), and each is an integer such that Ap-adic integer is ap-adic number such that
In general the series that represents ap-adic number is notconvergent in the usual sense, but it is convergent for thep-adic absolute value wherek is the least integeri such that (if all are zero, one has the zerop-adic number, which has0 as itsp-adic absolute value).
Every rational number can be uniquely expressed as the sum of a series as above, with respect to thep-adic absolute value. This allows considering rational numbers as specialp-adic numbers, and alternatively defining thep-adic numbers as thecompletion of the rational numbers for thep-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.
p-adic numbers were first described byKurt Hensel in 1897,[1] though, with hindsight, some ofErnst Kummer's earlier work can be interpreted as implicitly usingp-adic numbers.[note 2]
Roughly speaking,modular arithmetic modulo a positive integern consists of "approximating" every integer by the remainder of itsdivision byn, called itsresidue modulon. The main property of modular arithmetic is that the residue modulon of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulon.
When studyingDiophantine equations, it's sometimes useful to reduce the equation modulo a primep, since this usually provides more insight about the equation itself. Unfortunately, doing this loses some information because the reduction is not injective.
One way to preserve more information is to use larger moduli, such as higher prime powers,p2,p3, .... However, this has the disadvantage of not being a field, which loses a lot of the algebraic properties that has.[2]
Kurt Hensel discovered a method which consists of using a prime modulusp, and applyingHensel's lemma to lift solutions modulop to modulop2,p3, .... This process creates an infinite sequence of residues, and ap-adic number is defined as the "limit" of such a sequence.
Essentially,p-adic numbers allows "taking modulope for alle at once". A distinguishing feature ofp-adic numbers from ordinary modulo arithmetic is that the set ofp-adic numbers forms afield, making division byp possible (unlike when working modulope). Furthermore, the mapping isinjective, so not much information is lost when reducing top-adic numbers.[2]
There are multiple ways to understandp-adic numbers.
One way to think aboutp-adic integers is using "basep". For example, every integer can be written in basep,
Informally,p-adic integers can be thought of as integers in base-p, but the digits extendinfinitely to the left.[2]
Addition and multiplication onp-adic integers can be carried out similarly to integers in base-p.[3]
When adding together twop-adic integers, for example, their digits are added with carries being propagated from right to left.
Multiplication ofp-adic integers works similarly vialong multiplication. Since addition and multiplication can be performed withp-adic integers, they form aring, denoted or.
Note that some rational numbers can also bep-adic integers, even if they aren't integers in a real sense. For example, the rational number1/5 is a 3-adic integer, and has the 3-adic expansion. However, some rational numbers, such as, cannot be written as ap-adic integer. Because of this,p-adic integers are generalized further top-adic numbers:
p-adic numbers can be thought of asp-adic integers withfinitely many digits after the decimal point. An example of a 3-adic number is
Equivalently, everyp-adic number is of the form, wherex is ap-adic integer.
For anyp-adic numberx, itsmultiplicative inverse is also ap-adic number, which can be computed using a variant oflong division.[3] For this reason, thep-adic numbers form afield, denoted or.
Another way to definep-adic integers is by representing it as a sequence of residues mod for each integer,[2]
satisfying the compatibility relations for. In this notation, addition and multiplication ofp-adic integers are defined component-wise:
This is equivalent to the base-p definition, because the lastk digits of a base-p expansion uniquely define its value modpk, and vice versa.
This form can also explain why some rational numbers arep-adic integers, even if they are not integers. For example,1/5 is a 3-adic integer, because its 3-adic expansion consists of themultiplicative inverses of 5 mod 3, 32, 33, ...
There are several equivalent definitions ofp-adic numbers. The two approaches given below are relatively elementary.
Ap-adic integer is often defined as aformal power series of the formwhere each represents a "digit in basep".
Ap-adic unit is ap-adic integer whose first digit is nonzero, i.e.. The set of allp-adic integers is usually denoted.[4]
Ap-adic number is then defined as aformal Laurent series of the formwherev is a (possibly negative) integer, and each.[5] Equivalently, ap-adic number is anything of the form, wherex is ap-adic integer.
The first indexv for which the digit is nonzero inr is called thep-adic valuation ofr, denoted. If, then such an index does not exist, so by convention.
In this definition, addition, subtraction, multiplication, and division ofp-adic numbers are carried out similarly to numbers in basep, with "carries" or "borrows" moving from left to right rather than right to left.[6] As an example in,
Division ofp-adic numbers may also be carried out "formally" viadivision of formal power series, with some care about having to "carry".[5]
With these operations, the set ofp-adic numbers form afield, denoted.
Thep-adic numbers may also be defined as equivalence classes, in a similar way as the definition of real numbers as equivalence classes ofCauchy sequences. It is fundamentally based on the following lemma:
The exponentv is uniquely determined byr and is called itsp-adic valuation, denoted. The proof of the lemma results directly from thefundamental theorem of arithmetic.
Ap-adic series is aformal Laurent series of the formwhere is a (possibly negative) integer and the are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of is not divisible byp).
Every rational number may be viewed as ap-adic series with a single nonzero term, consisting of its factorization of the form withm andn both coprime withp.
Twop-adic series andareequivalent if there is an integerN such that, for every integer the rational numberis zero or has ap-adic valuation greater thann.
Ap-adic series isnormalized if either all are integers such that and or all are zero. In the latter case, the series is called thezero series.
Everyp-adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see§ Normalization of ap-adic series, below.
In other words, the equivalence ofp-adic series is anequivalence relation, and eachequivalence class contains exactly one normalizedp-adic series.
The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence ofp-adic series. That is, denoting the equivalence with~, ifS,T andU are nonzerop-adic series such that one has
With this, thep-adic numbers are defined as theequivalence classes ofp-adic series.
The uniqueness property of normalization, allows uniquely representing anyp-adic number by the corresponding normalizedp-adic series. The compatibility of the series equivalence leads almost immediately to basic properties ofp-adic numbers:
Starting with the series we wish to arrive at an equivalent series such that thep-adic valuation of is zero. For that, one considers the first nonzero If itsp-adic valuation is zero, it suffices to changev intoi, that is to start the summation fromv. Otherwise, thep-adic valuation of is and where the valuation of is zero; so, one gets an equivalent series by changing to0 and to Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation of is zero.
Then, if the series is not normalized, consider the first nonzero that is not an integer in the interval UsingBézout's lemma, write this as, where and has nonnegative valuation. Then, one gets an equivalent series by replacing with and adding to Iterating this process, possibly infinitely many times, provides eventually the desired normalizedp-adic series.
Other equivalent definitions usecompletion of adiscrete valuation ring (see§ p-adic integers),completion of a metric space (see§ Topological properties), orinverse limits (see§ Modular properties).
Ap-adic number can be defined as anormalizedp-adic series. Since there are other equivalent definitions that are commonly used, one says often that a normalizedp-adic seriesrepresents ap-adic number, instead of saying that itis ap-adic number.
One can say also that anyp-adic series represents ap-adic number, since everyp-adic series is equivalent to a unique normalizedp-adic series. This is useful for defining operations (addition, subtraction, multiplication, division) ofp-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations onp-adic numbers, since the series operations are compatible with equivalence ofp-adic series.
With these operations,p-adic numbers form afield called thefield ofp-adic numbers and denoted or There is a uniquefield homomorphism from the rational numbers into thep-adic numbers, which maps a rational number to itsp-adic expansion. Theimage of this homomorphism is commonly identified with the field of rational numbers. This allows considering thep-adic numbers as anextension field of the rational numbers, and the rational numbers as asubfield of thep-adic numbers.
Thevaluation of a nonzerop-adic numberx, commonly denoted is the exponent ofp in the first nonzero term of everyp-adic series that representsx. By convention, that is, the valuation of zero is This valuation is adiscrete valuation. The restriction of this valuation to the rational numbers is thep-adic valuation of that is, the exponentv in the factorization of a rational number as with bothn anddcoprime withp.
There are several different conventions for writingp-adic expansions. So far this article has used a notation forp-adic expansions in whichpowers ofp increase from right to left. With this right-to-left notation the 3-adic expansion of for example, is written as
When performing arithmetic in this notation, digits arecarried to the left. It is also possible to writep-adic expansions so that the powers ofp increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of is
p-adic expansions may be written withother sets of digits instead of{0, 1, ..., p − 1}. For example, the3-adic expansion of can be written usingbalanced ternary digits{1, 0, 1}, with1 representing negative one, as
In fact any set ofp integers which are in distinctresidue classes modulop may be used asp-adic digits. In number theory,Teichmüller representatives are sometimes used as digits.[7]
Quote notation is a variant of thep-adic representation ofrational numbers that was proposed in 1979 byEric Hehner andNigel Horspool for implementing on computers the (exact) arithmetic with these numbers.[8] It can be used as a compact way to represent rational numbers, which have an infinite periodic sequence of digits. In this notation, a quote mark (') is used to separate the repeating part from the nonrepeating part.
Thedecimal expansion of a positiverational number is its representation as aserieswhere is an integer and each is also aninteger such that This expansion can be computed bylong division of the numerator by the denominator, which is itself based on the following theorem: If is a rational number such that there is an integer such that and with The decimal expansion is obtained by repeatedly applying this result to the remainder which in the iteration assumes the role of the original rational number.
Thep-adic expansion of a rational number can be computed similarly, but with a different division step. Suppose that is a rational number with nonnegative valuation (that is,d is not divisible byp). The division step consists of writingwhere is an integer such that and has nonnegative valuation.
The integera can be computed as amodular multiplicative inverse:. Because of this, writingr in this way is always possible, and such a representation is unique.
Thep-adic expansion of a rational number is eventuallyperiodic.Conversely, a series with converges (for thep-adic absolute value) to a rational numberif and only if it is eventually periodic; in this case, the series is thep-adic expansion of that rational number. Theproof is similar to that of the similar result forrepeating decimals.
Let us compute the 5-adic expansion of We can write this number as. Thus we use for the first step.For the next step, we can write the "remainder" as. Thus we use.We can write the "remainder" as. Thus we use.Notice that we obtain the "remainder" again, which means the digits can only repeat from this point on.In the standard 5-adic notation, we can write this aswith theellipsis on the left hand side.
Thep-adic integers are thep-adic numbers with a nonnegative valuation.
A-adic integer can be represented as a sequenceof residues mod for each integer, satisfying the compatibility relations for.
Everyinteger is a-adic integer (including zero, since). The rational numbers of the form with coprime with and are also-adic integers (for the reason that has an inverse mod for every).
Thep-adic integers form acommutative ring, denoted or, that has the following properties.
The last property provides a definition of thep-adic numbers that is equivalent to the above one: the field of thep-adic numbers is thefield of fractions of the completion of the localization of the integers at the prime ideal generated byp.
Thep-adic valuation allows defining anabsolute value onp-adic numbers: thep-adic absolute value of a nonzerop-adic numberx iswhere is thep-adic valuation ofx. Thep-adic absolute value of is This is an absolute value that satisfies thestrong triangle inequality since, for everyx andy:
Moreover, if then
This makes thep-adic numbers ametric space, and even anultrametric space, with thep-adic distance defined by
As a metric space, thep-adic numbers form thecompletion of the rational numbers equipped with thep-adic absolute value. This provides another way for defining thep-adic numbers.
As the metric is defined from adiscrete valuation, everyopen ball is alsoclosed. More precisely, the open ball equals the closed ball wherev is the least integer such that Similarly, wherew is the greatest integer such that
This implies that thep-adic numbers form alocally compact space (locally compact field), and thep-adic integers—that is, the ball—form acompact space.[9]
The space of 2-adic integers ishomeomorphic to theCantor set.[10][11] This can be seen by considering the continuous 1-to-1 mapping defined byMoreover, for anyp, is homeomorphic to, and therefore also homeomorphic to the Cantor set.[12]
ThePontryagin dual of the group ofp-adic integers is thePrüferp-group, and the Pontryagin dual of the Prüferp-group is the group ofp-adic integers.[13]
Thequotient ring may be identified with thering of the integersmodulo This can be shown by remarking that everyp-adic integer, represented by its normalizedp-adic series, is congruent modulo with itspartial sum whose value is an integer in the interval A straightforward verification shows that this defines aring isomorphism from to
Theinverse limit of the rings is defined as the ring formed by the sequences such that and for everyi.
The mapping that maps a normalizedp-adic series to the sequence of its partial sums is a ring isomorphism from to the inverse limit of the This provides another way for definingp-adic integers (up to an isomorphism).
This definition ofp-adic integers is specially useful for practical computations, as allowing buildingp-adic integers by successive approximations.
For example, for computing thep-adic (multiplicative) inverse of an integer, one can useNewton's method, starting from the inverse modulop; then, each Newton step computes the inverse modulo from the inverse modulo
The same method can be used for computing thep-adicsquare root of an integer that is aquadratic residue modulop. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found in. Applying Newton's method to find the square root requires to be larger than twice the given integer, which is quickly satisfied.
Hensel lifting is a similar method that allows to "lift" the factorization modulop of a polynomial with integer coefficients to a factorization modulo for large values ofn. This is commonly used bypolynomial factorization algorithms.
Both and areuncountable and have thecardinality of the continuum.[14] For this results from thep-adic representation, which defines abijection of on thepower set For this results from its expression as acountably infiniteunion of copies of:
contains and is a field ofcharacteristic0.
Because0 can be written as sum of squares,[note 3] cannot be turned into anordered field.
The field ofreal numbers has only a single properalgebraic extension: thecomplex numbers. In other words, thisquadratic extension is alreadyalgebraically closed. By contrast, thealgebraic closure of, denoted has infinite degree,[15] that is, has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of thep-adic valuation to the latter is not (metrically) complete.[16][17]
Its (metric) completion is denoted or,[17][18] and sometimes called thecomplexp-adic numbers by analogy to the complex numbers. Here an end is reached, as is algebraically closed.[17][19] However unlike this field is notlocally compact.[18]
and are isomorphic as rings,[note 4] so we may regard as endowed with an exotic metric. The proof of existence of such a field isomorphism relies on theaxiom of choice, and does not provide an explicit example of such an isomorphism (that is, it is notconstructive).
If is any finiteGalois extension of theGalois group issolvable. Thus, the Galois group isprosolvable.
contains then-thcyclotomic field (n > 2) if and only ifn |p − 1.[20] For instance, then-th cyclotomic field is a subfield of if and only ifn = 1, 2, 3, 4, 6, or12. In particular, there is no multiplicativep-torsion in ifp > 2. Also,−1 is the only non-trivial torsion element in.
Given anatural numberk, theindex of the multiplicative group of thek-th powers of the non-zero elements of in is finite.
The numbere, defined as the sum ofreciprocals offactorials, is not a member of anyp-adic field; but for. Forp = 2 one must take at least the fourth power.[21] (Thus a number with similar properties ase — namely ap-th root ofep — is a member of for allp.)
Helmut Hasse'slocal–global principle is said to hold for an equation if it can be solved over the rational numbersif and only if it can be solved over the real numbers and over thep-adic numbers for every prime p. This principle holds, for example, for equations given byquadratic forms, but fails for higher polynomials in several indeterminates.
Thep-adic numbers have appeared in several fields of mathematics as well as physics.
Similar to the more classical fields ofreal andcomplex analysis, which deal, respectively, with functions on the real and complex numbers,p-adic analysis studies functions onp-adic numbers. The theory of complex-valued numerical functions on thep-adic numbers is part of the theory oflocally compact groups (abstract harmonic analysis). The usual meaning taken forp-adic analysis is the theory ofp-adic-valued functions on spaces of interest.
Applications ofp-adic analysis have mainly been in number theory, where it has a significant role indiophantine geometry anddiophantine approximation. Some applications have required the development ofp-adicfunctional analysis andspectral theory. In many waysp-adic analysis is less subtle thanclassical analysis, since theultrametric inequality means, for example, that convergence ofinfinite series ofp-adic numbers is much simpler.Topological vector spaces overp-adic fields show distinctive features; for example aspects relating toconvexity and theHahn–Banach theorem are different.
Two important concepts fromp-adic analysis areMahler's theorem, which characterizes every continuousp-adic function in terms of polynomials, andVolkenborn integral, which provides a method ofintegration forp-adic functions.
p-adic Hodge theory is a theory that provides a way to classify and studyp-adic Galois representations ofcharacteristic 0local fields with residual characteristicp (such asQp). The theory has its beginnings inJean-Pierre Serre andJohn Tate's study ofTate modules ofabelian varieties and the notion ofHodge–Tate representation. Hodge–Tate representations are related to certain decompositions ofp-adiccohomology theories analogous to theHodge decomposition, hence the namep-adic Hodge theory. Further developments were inspired by properties ofp-adic Galois representations arising from theétale cohomology ofvarieties.Jean-Marc Fontaine introduced many of the basic concepts of the field.
p-adic Teichmüller theory describes the "uniformization" ofp-adic curves and theirmoduli, generalizing the usualTeichmüller theory that describes theuniformization ofRiemann surfaces and their moduli. It was introduced and developed byShinichi Mochizuki.
p-adic quantum mechanics is a collection of related research efforts inquantum physics that replace real numbers withp-adic numbers. Historically, this research was inspired by the discovery that theVeneziano amplitude of the openbosonic string, which is calculated using an integral over the real numbers, can be generalized to thep-adic numbers. This observation initiated the study ofp-adic string theory.
The reals and thep-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance generalalgebraic number fields, in an analogous way. This will be described now.
SupposeD is aDedekind domain andE is itsfield of fractions. Pick a non-zeroprime idealP ofD. Ifx is a non-zero element ofE, thenxD is afractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals ofD. We write ordP(x) for the exponent ofP in this factorization, and for any choice of numberc greater than 1 we can setCompleting with respect to this absolute value|⋅|P yields a fieldEP, the proper generalization of the field ofp-adic numbers to this setting. The choice ofc does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when theresidue fieldD/P is finite, to take forc the size ofD/P.
For example, whenE is anumber field,Ostrowski's theorem says that every non-trivialnon-Archimedean absolute value onE arises as some|⋅|P. The remaining non-trivial absolute values onE arise from the different embeddings ofE into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings ofE into the fieldsCp, thus putting the description of allthe 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 whenE is a number field (or more generally aglobal field), which are seen as encoding "local" information. This is accomplished byadele rings andidele groups.
p-adic integers can be extended top-adic solenoids. There is a map from to thecircle group whose fibers are thep-adic integers, in analogy to how there is a map from to the circle whose fibers are.
Thep-adic integers can also be extended toprofinite integers, which can be understood as thedirect product of ringsUnlike thep-adic integers which only generalize the modulo over prime powerspk, the profinite integers generalizes the modulo overall natural numbersn.
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