P-adic number

thumb|upright=1.2|The 3-adic integers, depicted as a slice of a [[solenoid (mathematics)|solenoid.]]

In number theory, given a prime number , the -adic numbers form an extension of the rational numbers that is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right.

For example, comparing the expansion of the rational number <math>\tfrac15</math> in base vs. the -adic expansion,

<math display="block">\begin{alignat}{3}

\tfrac15 &{}= 0.01210121\ldots \ (\text{base } 3)

&&{}= 0\cdot 3^0 + 0\cdot 3^{-1} + 1\cdot 3^{-2} + 2\cdot 3^{-3} + \cdots \\[5mu]

\tfrac15 &{}= \dots 121012102 \ \ (\text{3-adic})

&&{}= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0.

\end{alignat}</math>

Formally, given a prime number , a -adic number can be defined as a series

<math display="block">s=\sum_{i=k}^\infty a_i p^i = a_k p^k + a_{k+1} p^{k+1} + a_{k+2} p^{k+2} + \cdots</math>

where is an integer (possibly negative), and each <math>a_i</math> is an integer such that <math>0\le a_i < p.</math> A -adic integer is a -adic number such that <math>k\ge 0.</math>

In general the series that represents a -adic number is not convergent in the usual sense, but it is convergent for the -adic absolute value <math>|s|_p=p^{-k},</math> where is the least integer such that <math>a_i\ne 0</math> (if all <math>a_i</math> are zero, one has the zero -adic number, which has as its -adic absolute value).

Every rational number can be uniquely expressed as the sum of a series as above, with respect to the -adic absolute value. This allows considering rational numbers as special -adic numbers, and alternatively defining the -adic numbers as the completion of the rational numbers for the -adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.

-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.

Motivation

Roughly speaking, modular arithmetic modulo a positive integer consists of "approximating" every integer by the remainder of its division by , called its residue modulo . The main property of modular arithmetic is that the residue modulo of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo .

When studying Diophantine equations, it's sometimes useful to reduce the equation modulo a prime , since this usually provides more insight about the equation itself. Unfortunately, doing this loses some information because the reduction <math>\mathbb Z\twoheadrightarrow\mathbb Z/p</math> is not injective.

One way to preserve more information is to use larger moduli, such as higher prime powers, , . However, this has the disadvantage of <math>\mathbb Z/p^e</math> not being a field, which loses a lot of the algebraic properties that <math>\mathbb Z/p</math> has.

Kurt Hensel discovered a method which consists of using a prime modulus , and applying Hensel's lemma to lift solutions modulo to modulo , . This process creates an infinite sequence of residues, and a -adic number is defined as the "limit" of such a sequence.

Essentially, -adic numbers allows "taking modulo for all at once". A distinguishing feature of -adic numbers from ordinary modulo arithmetic is that the set of -adic numbers <math>\mathbb Q_p</math> forms a field, making division by possible (unlike when working modulo ). Furthermore, the mapping <math>\mathbb Z\hookrightarrow\mathbb Z_p</math> is injective, so not much information is lost when reducing to -adic numbers.

When adding together two -adic integers, for example <math>\ldots012102_3+\ldots101211_3</math>, their digits are added with carries being propagated from right to left.

<math display="block">\begin{array}{cccccccc}

& & &_1 &_1 & &_1 & \\

& \cdots & 0 & 1 & 2 & 1 & 0 & 2 \,_3 \\

+ & \cdots & 1 & 0 & 1 & 2 & 1 & 1 \,_3 \\\hline

& \cdots & 1 & 2 & 1 & 0 & 2 & 0 \,_3

\end{array}</math>

Multiplication of -adic integers works similarly via long multiplication. Since addition and multiplication can be performed with -adic integers, they form a ring, denoted <math>\mathbb Z_p</math> or <math>\mathbf Z_p</math>.

Note that some rational numbers can also be -adic integers, even if they aren't integers in a real sense. For example, the rational number is a 3-adic integer, and has the 3-adic expansion <math>\tfrac{1}{5}=\ldots121012102_3</math>. However, some rational numbers, such as <math>\tfrac{1}{p}</math>, cannot be written as a -adic integer. Because of this, -adic integers are generalized further to -adic numbers:

-adic numbers can be thought of as -adic integers with finitely many digits after the decimal point. An example of a 3-adic number is

<math display="block">\ldots121012.102_3 = \cdots+1\cdot3^1+2\cdot3^0+1\cdot3^{-1}+0\cdot3^{-2}+2\cdot3^{-3}</math>

Equivalently, every -adic number is of the form <math>\tfrac x{p^k}</math>, where is a -adic integer.

For any nonzero -adic number , its multiplicative inverse <math>\tfrac{1}{x}</math> is also a -adic number, which can be computed using a variant of long division.

A -adic number is then defined as a formal Laurent series of the form

<math display="block">r=\sum_{i=v}^\infty a_i p^i = a_v p^v + a_{v+1} p^{v+1} + a_{v+2} p^{v+2} + a_{v+3} p^{v+3} + \cdots</math>

where is a (possibly negative) integer, and each <math>a_i\in\{0,1,\ldots,p-1\}</math>. Equivalently, a -adic number is anything of the form <math>\tfrac{x}{p^k}</math>, where is a -adic integer.

The first index for which the digit <math>a_v</math> is nonzero in is called the -adic valuation of , denoted <math>v_p(r)</math>. If <math>r=0</math>, then such an index does not exist, so by convention <math>v_p(0)=\infty</math>.

In this definition, addition, subtraction, multiplication, and division of -adic numbers are carried out similarly to numbers in base , with "carries" or "borrows" moving from left to right rather than right to left. As an example in <math>\mathbb Q_3</math>,

<math display="block">\begin{array}{lllllllllll}

& & & _1 & & & & _1 & & _1 \\

& 2\cdot3^0 &+& 0\cdot3^1 &+& 1\cdot3^2 &+& 2\cdot3^3 &+& 1\cdot3^4 &+ \cdots \\

+ & 1\cdot3^0 &+& 1\cdot3^1 &+& 2\cdot3^2 &+& 1\cdot3^3 &+& 0\cdot3^4 &+ \cdots \\\hline

& 0\cdot3^0 &+& 2\cdot3^1 &+& 0\cdot3^2 &+& 1\cdot3^3 &+& 2\cdot3^4 &+ \cdots

\end{array}</math>

Division of -adic numbers may also be carried out "formally" via division of formal power series, with some care about having to "carry".

is a variant of the -adic representation of rational numbers that was proposed in 1979 by Eric Hehner and Nigel Horspool for implementing on computers the (exact) arithmetic with these numbers. 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.

p-adic expansion of rational numbers

The decimal expansion of a positive rational number <math>r</math> is its representation as a series

<math display="block">r = \sum_{i=k}^\infty a_i 10^{-i},</math>

where <math>k</math> is an integer and each <math>a_i</math> is also an integer such that <math>0\le a_i <10.</math> This expansion can be computed by long division of the numerator by the denominator, which is itself based on the following theorem: If <math>r=\tfrac n d</math> is a rational number such that <math>0\le r <1,</math> there is an integer <math>a</math> such that <math>0\le a <10,</math> and <math>10r = a+r',</math> with <math>0\le r'<1.</math> The decimal expansion is obtained by repeatedly applying this result to the remainder <math>r'</math> which in the iteration assumes the role of the original rational number <math>r</math>.

The -adic expansion of a rational number can be computed similarly, but with a different division step. Suppose that <math>r=\tfrac{n}{d}</math> is a rational number with nonnegative valuation (that is, is not divisible by ). The division step consists of writing

where <math>a</math> is an integer such that <math>0\le a <p,</math> and <math>r'</math> has nonnegative valuation.

The integer can be computed as a modular multiplicative inverse: <math>a=nd^{-1}\operatorname{mod}p</math>. Because of this, writing in this way is always possible, and such a representation is unique.

The -adic expansion of a rational number is eventually periodic. Conversely, a series <math display=inline>\sum_{i=k}^\infty a_i p^i,</math> with <math>0\le a_i <p</math> converges (for the -adic absolute value) to a rational number if and only if it is eventually periodic; in this case, the series is the -adic expansion of that rational number. The proof is similar to that of the similar result for repeating decimals.

Example

Let us compute the 5-adic expansion of <math>\tfrac 13.</math> We can write this number as <math>\tfrac13 = 2 + 5 \cdot \tfrac{-1}3</math>. Thus we use <math>a=2</math> for the first step.

<math display="block">\frac13 = 2 + 5^1 \cdot \left(\frac{-1}3\right)</math>

For the next step, we can write the "remainder" <math>\tfrac{-1}3</math> as <math>\tfrac{-1}3 = 3 + 5 \cdot \tfrac{-2}3</math>. Thus we use <math>a=3</math>.

<math display="block">\frac13 = 2 + 3\cdot 5^1 + 5^2 \cdot \left(\frac{-2}3\right)</math>

We can write the "remainder" <math>\tfrac{-2}3</math> as <math>\tfrac{-2}3 = 1 + 5 \cdot \tfrac{-1}3</math>. Thus we use <math>a=1</math>.

<math display="block">\frac13 = 2 + 3\cdot 5^1 + 1\cdot5^2 + 5^3\cdot \left(\frac{-1}3\right)</math>

Notice that we obtain the "remainder" <math>\tfrac{-1}3</math> again, which means the digits can only repeat from this point on.

<math display="block">\frac13 = 2 + 3\cdot 5^1 + 1\cdot5^2 + 3\cdot 5^3 + 1\cdot 5^4 + 3\cdot 5^5 + 1\cdot 5^6 + \cdots</math>

In the standard 5-adic notation, we can write this as

<math display="block">\frac 13= \ldots 1313132_5 </math>

with the ellipsis <math> \ldots </math> on the left hand side.

p-adic integers

The -adic integers are the -adic numbers with a nonnegative valuation.

A <math>p</math>-adic integer can be represented as a sequence

<math display="block"> x = (x_1 \operatorname{mod} p, ~ x_2 \operatorname{mod} p^2, ~ x_3 \operatorname{mod} p^3, ~ \ldots)</math>

of residues <math>x_e</math> mod <math>p^e</math> for each integer <math>e</math>, satisfying the compatibility relations <math>x_i \equiv x_j ~ (\operatorname{mod} p^i)</math> for <math>i < j</math>.

Every integer is a <math>p</math>-adic integer (including zero, since <math>0<\infty</math>). The rational numbers of the form <math display=inline> \tfrac nd p^k</math> with <math>d</math> coprime with <math>p</math> and <math>k\ge 0</math> are also <math>p</math>-adic integers (for the reason that <math>d</math> has an inverse mod <math>p^e</math> for every <math>e</math>).

The -adic integers form a commutative ring, denoted <math>\Z_p</math> or <math>\mathbf Z_p</math>, that has the following properties.

It is an integral domain, since it is a subring of a field, or since the first term of the series representation of the product of two non zero -adic series is the product of their first terms.
The units (invertible elements) of <math>\Z_p</math> are the -adic numbers of valuation zero.
It is a principal ideal domain, such that each ideal is generated by a power of .
It is a local ring of Krull dimension one, since its only prime ideals are the zero ideal and the ideal generated by , the unique maximal ideal.
It is a discrete valuation ring, since this results from the preceding properties.
It is the completion of the local ring <math>\Z_{(p)} = \bigl\{\tfrac nd \mathbin{\big|} n, d \in \Z,\, d \not\in p\Z \bigr\},</math> which is the localization of <math>\Z</math> at the prime ideal <math>p\Z.</math>

The last property provides a definition of the -adic numbers that is equivalent to the above one: the field of the -adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by .

Topological properties

thumb|Visual depiction of the 3-adic metric applied to the , the integers modulo 27

The -adic valuation allows defining an absolute value on -adic numbers: the -adic absolute value of a nonzero -adic number is

where <math>v_p(x)</math> is the -adic valuation of . The -adic absolute value of <math>0</math> is <math>|0|_p = 0.</math> This is an absolute value that satisfies the strong triangle inequality since, for every and :

<math>|x|_p = 0</math> if and only if <math>x=0;</math>
<math>|x|_p\cdot |y|_p = |xy|_p;</math>
<math>|x+y|_p\le \max\bigl(|x|_p,|y|_p\bigr) \le |x|_p + |y|_p.</math>

Moreover, if <math>|x|_p \ne |y|_p,</math> then <math>|x+y|_p = \max\bigl(|x|_p,|y|_p\bigr).</math>

This makes the -adic numbers a metric space, and even an ultrametric space, with the -adic distance defined by

As a metric space, the -adic numbers form the completion of the rational numbers equipped with the -adic absolute value. This provides another way for defining the -adic numbers.

As the metric is defined from a discrete valuation, every open ball is also closed. More precisely, the open ball <math>B_r(x) =\{y\mid d_p(x,y)<r\}</math> equals the closed ball <math>\textstyle B_{p^{-v[x] =\{y\mid d_p(x,y)\le p^{-v}\},</math> where is the least integer such that <math>\textstyle p^{-v}< r.</math> Similarly, <math>\textstyle B_r[x] = B_{p^{-w(x),</math> where is the greatest integer such that <math>\textstyle p^{-w}>r.</math>

This implies that the -adic numbers <math>\mathbb Q_p</math> form a locally compact space (locally compact field), and the -adic integers <math>\mathbb Z_p</math>—that is, the ball <math>B_1[0]=B_p(0)</math>—form a compact space.

The space of 2-adic integers <math>\mathbb Z_2</math> is homeomorphic to the Cantor set <math>\mathcal C</math>. This can be seen by considering the continuous 1-to-1 mapping <math>\psi:\mathbb Z_2\to \mathcal C</math> defined by

<math display="block">\psi:~a_0+a_12+a_22^2+a_32^3+\cdots~\longmapsto~\frac{2a_0}3+\frac{2a_1}{3^2}+\frac{2a_2}{3^3}+\frac{2a_3}{3^4}+\cdots</math>

Moreover, for any , <math>\mathbb Z_p</math> is homeomorphic to <math>\mathbb Z_2</math>, and therefore also homeomorphic to the Cantor set.

The Pontryagin dual of the group of -adic integers is the Prüfer -group <math>\mathbb Z(p^\infty)</math>, and the Pontryagin dual of the Prüfer -group is the group of -adic integers.

Modular properties

The quotient ring <math>\Z_p/p^n\Z_p</math> may be identified with the ring <math>\Z/p^n\Z</math> of the integers modulo <math>p^n.</math> This can be shown by remarking that every -adic integer, represented by its normalized -adic series, is congruent modulo <math>p^n</math> with its partial sum <math display = inline>\sum_{i=0}^{n-1}a_ip^i,</math> whose value is an integer in the interval <math>[0,p^n-1].</math> A straightforward verification shows that this defines a ring isomorphism from <math>\Z_p/p^n\Z_p</math> to <math>\Z/p^n\Z.</math>

The inverse limit of the rings <math>\Z_p/p^n\Z_p</math> is defined as the ring formed by the sequences <math>a_0, a_1, \ldots</math> such that <math>a_i \in \Z/p^i \Z</math> and <math display = inline>a_i \equiv a_{i+1} \pmod {p^i}</math> for every .

The mapping that maps a normalized -adic series to the sequence of its partial sums is a ring isomorphism from <math>\Z_p</math> to the inverse limit of the <math>\Z_p/p^n\Z_p.</math> This provides another way for defining -adic integers (up to an isomorphism).

This definition of -adic integers is specially useful for practical computations, as allowing building -adic integers by successive approximations.

For example, for computing the -adic (multiplicative) inverse of an integer, one can use Newton's method, starting from the inverse modulo ; then, each Newton step computes the inverse modulo <math display = inline>p^{n^2}</math> from the inverse modulo <math display = inline>p^n.</math>

The same method can be used for computing the -adic square root of an integer that is a quadratic residue modulo . 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 <math>\Z_p/p^n\Z_p</math>. Applying Newton's method to find the square root requires <math display = inline>p^n</math> to be larger than twice the given integer, which is quickly satisfied.

Hensel lifting is a similar method that allows to "lift" the factorization modulo of a polynomial with integer coefficients to a factorization modulo <math display = inline>p^n</math> for large values of . This is commonly used by polynomial factorization algorithms.

Cardinality

Both <math>\Z_p</math> and <math>\Q_p</math> are uncountable and have the cardinality of the continuum. For <math>\Z_p,</math> this results from the -adic representation, which defines a bijection of <math>\Z_p</math> on the power set <math>\{0,\ldots,p-1\}^\N.</math> For <math>\Q_p</math> this results from its expression as a countably infinite union of copies of <math>\Z_p</math>:

<math display="block">\Q_p=\bigcup_{i=0}^\infty \frac 1{p^i}\Z_p.</math>

Algebraic closure

The field <math>\Q_p</math> contains <math>\Q</math> and is a field of characteristic .

Because can be written as sum of squares,<math>\Q_p</math> cannot be turned into an ordered field.

The field of real numbers <math>\R</math> has only a single proper algebraic extension: the complex numbers <math>\C</math>. In other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of <math>\Q_p</math>, denoted <math>\overline{\Q_p},</math> has infinite degree, that is, <math>\Q_p</math> has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the -adic valuation to <math>\overline{\Q_p},</math> the latter is not (metrically) complete.

Its (metric) completion is denoted <math>\C_p</math> or <math>\Omega_p</math>, and sometimes called the complex -adic numbers by analogy to the complex numbers. The field <math>\C_p</math> is algebraically closed. However, unlike <math>\C</math>, this field is not locally compact.

Given a natural number , let <math>\Q_p^{\times k}</math> be the group of kth powers of elements of <math>\Q_p^\times</math>; then the index <math>(\Q_p^\times:\Q_p^{\times k})</math> is finite.

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 -adic numbers for every prime . This principle holds, for example, for equations given by quadratic forms, but fails for higher polynomials in several indeterminates.

Rational arithmetic with Hensel lifting

Applications

The p-adic numbers have appeared in several fields of mathematics as well as physics.

Analysis

Similar to the more classical fields of real and complex analysis, which deal, respectively, with functions on the real and complex numbers, p-adic analysis studies functions on p-adic numbers. The theory of complex-valued numerical functions on the p-adic numbers is part of the theory of locally compact groups (abstract harmonic analysis). The usual meaning taken for p-adic analysis is the theory of p-adic-valued functions on spaces of interest.

Applications of p-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of p-adic functional analysis and spectral theory. In many ways p-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers is much simpler. Topological vector spaces over p-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theorem are different.

Two important concepts from p-adic analysis are Mahler's theorem, which characterizes every continuous p-adic function in terms of polynomials, and Volkenborn integral, which provides a method of integration for p-adic functions.

Hodge theory

p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.

Teichmüller theory

p-adic Teichmüller theory describes the "uniformization" of p-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfaces and their moduli. It was introduced and developed by Shinichi Mochizuki.

Quantum physics

p-adic quantum mechanics is a collection of related research efforts in quantum physics that replace real numbers with p-adic numbers. Historically, this research was inspired by the discovery that the Veneziano amplitude of the open bosonic string, which is calculated using an integral over the real numbers, can be generalized to the p-adic numbers. This observation initiated the study of p-adic string theory.

The reals and the -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. Therefore, writing ordP(x) for the exponent of P in this factorization gives a well-defined discrete valuation, and for any choice of number c greater than 1 we can set

<math display="block">|x|_P = c^{-\!\operatorname{ord}_P(x)}.</math>

Completing with respect to this absolute value |⋅|P yields a field EP, 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 and D is its ring of integers, 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 Cp, 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.

p-adic integers can be extended to p-adic solenoids <math>\mathbb{T}_p</math>. There is a map from <math>\mathbb{T}_p</math> to the circle group whose fibers are the p-adic integers <math>\mathbb{Z}_p</math>, in analogy to how there is a map from <math>\mathbb{R}</math> to the circle whose fibers are <math>\mathbb{Z}</math>.

The p-adic integers can also be extended to profinite integers <math>\widehat{\mathbb{Z</math>, which can be understood as the direct product of rings

<math display="block">\widehat{\mathbb{Z = \prod_p \mathbb{Z}_p.</math>

Unlike the p-adic integers which only generalize the modulo over prime powers pk, the profinite integers generalizes the modulo over all natural numbers n. The Chinese remainder theorem likewise implies a structure of <math>\mathbb{Z}_n</math> for composite bases: for any n which has at least two distinct prime factors, the n-adic integer ring <math>\mathbb{Z}_n</math> is isomorphic to <math>\prod_{p | n} \mathbb{Z}_p</math>.

Footnotes

Notes

Citations

References

. — Translation into English by John Stillwell of Theorie der algebraischen Functionen einer Veränderlichen (1882).

External links

p-adic number at Springer On-line Encyclopaedia of Mathematics

Motivation

p-adic expansion of rational numbers

Example

p-adic integers

Topological properties

Modular properties

Cardinality

Algebraic closure <span class="anchor" id="Complex p-adic number"></span>

Local–global principle

Rational arithmetic with Hensel lifting

Applications

Analysis

Hodge theory

Teichmüller theory

Quantum physics

See also

Footnotes

Notes

Citations

References

Further reading

External links

Motivation

p-adic expansion of rational numbers

Example

p-adic integers

Topological properties

Modular properties

Cardinality

Algebraic closure <span class="anchor" id="Complex p-adic number"></span>

Local–global principle

Rational arithmetic with Hensel lifting

Applications

Analysis

Hodge theory

Teichmüller theory

Quantum physics

Generalizations and related concepts

See also

Footnotes

Notes

Citations

References

Further reading

External links