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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the -adic number system for any
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
  extends the ordinary
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
of the
rational numbers In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
in a different way from the extension of the rational
number system A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduc ...
to the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
and
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode
congruence Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
information in a way that turns out to have powerful applications in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
– including, for example, in the famous proof of
Fermat's Last Theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
by
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national a ...
. These numbers were first described by
Kurt Hensel #REDIRECT Kurt Hensel Kurt Wilhelm Sebastian Hensel (29 December 1861 – 1 June 1941) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes ...
in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introduction
page 35
"Indeed, with hindsight it becomes apparent that a discrete valuation is behind Kummer's concept of ideal numbers."
The -adic numbers were motivated primarily by an attempt to bring the ideas and techniques of
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
methods into
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
. Their influence now extends far beyond this. For example, the field of -adic analysis essentially provides an alternative form of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
. More formally, for a given prime , the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
of -adic numbers is a completion of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s. The field is also given a
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
derived from a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geomet ...
, which is itself derived from the -adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every
Cauchy sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
converges to a point in . This is what allows the development of calculus on , and it is the interaction of this analytic and
algebraic
algebraic
structure that gives the -adic number systems their power and utility. The in "-adic" is a variable and may be replaced with a prime (yielding, for instance, "the 2-adic numbers") or another expression representing a prime number. The "adic" of "-adic" comes from the ending found in words such as
dyadic Dyadic describes the interaction between two things, and may refer to: *Dyad (sociology), interaction between a pair of individuals *Dyadic counterpoint, the voice-against-voice conception of polyphony *People who are not intersex (see also endose ...
or triadic.


''p''-adic expansion of rational numbers

The
decimal expansion A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive ...
of a positive
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
is its representation as a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
:r = \sum_^\infty a_i 10^, where each a_i is an
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
such that 0\le a_i <10. This expansion can be computed by
long division In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

long division
of the numerator by the denominator, which is itself based on the following theorem: If r=\tfrac nd is a rational number such that 10^k\le r <10^, there is an integer such that 0< a <10, and r = a\,10^k +s, with s<10^k. 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 . The -''adic expansion'' of a rational number is defined similarly, but with a different division step. More precisely, given a fixed
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, every nonzero
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
r can be uniquely written as r=p^k\tfrac nd, where is a (possibly negative) integer, and and are coprime integers both coprime with . The integer is the -adic valuation of , denoted v_p(r), and p^ is its -adic absolute value, denoted , r, _p (the absolute value is small when the valuation is large). The division step consists of writing :r = a\,p^k + s where is an integer such that 0\le a and is either zero, or a rational number such that , s, _p < p^ (that is, v_p(s)>k). The -''adic expansion'' of is the
formal power series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
:r = \sum_^\infty a_ip^i obtained by repeating indefinitely the division step on successive remainders. In a -adic expansion, all a_i are integers such that 0\le a_i If r=p^k \tfrac n1 with , the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation of in base . The existence and the computation of the -adic expansion of a rational number result of
Bézout's identity In elementary number theory, Bézout's identity (also called Bézout's lemma) is the following theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical ...
in the following way. If, as above, r=p^k \tfrac nd, and and are coprime, there exist integers and such that td+up=1. So :r=p^k \tfrac nd(td+up)=p^knt + p^\frac d. Then, the
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
of by gives :nt=qp+a, with 0\le a This gives the division step as :\begin r & = & p^k(qp+a) + p^\frac d \\ & = & ap^k +p^\,\frac d, \\ \end so that in the iteration :s= p^\,\frac d is the new rational number. The uniqueness of the division step and of the whole -adic expansion is easy: if p^ka + p^s=p^ka' + p^s', one has a-a'=p(s'-s). This means divides a-a'. Since 0\le a and 0\le a' the following must be true: 0\le a and a' Thus, one gets -p Since divides a-a', this proves a=a'. The -adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a
convergent series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
with the -adic absolute value. In the standard -adic notation, the digits are written in the same order as in a standard base- system, namely with the powers of the base increasing to the left. This means that the production of the digits is reversed and the limit happens on the left hand side. The -adic expansion of a rational number is eventually
periodic
periodic
. Conversely, a series \sum_^\infty a_ip^i, with 0\le a_i 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 decimal A repeating decimal or recurring decimal is decimal representation of a number whose Numerical digit, digits are periodic function, periodic (repeating its values at regular intervals) and the infinity, infinitely repeated portion is not zero. It c ...
s.


Example

Let us compute the 5-adic expansion of \frac 13. Bézout's identity for 5 and the denominator 3 is 2\cdot 3 + (-1)\cdot 5 =1 (for larger examples, this can be computed with the
extended Euclidean algorithm In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' ...
). Thus :\frac 13= 2-\frac 53. For the next step, one has to "divide" -1/3 (the factor 5 in the numerator of the fraction has to be viewed as a " shift" of the -adic valuation, and thus it is not involved in the "division"). Multiplying Bézout's identity by -1 gives :-\frac 13=-2+\frac 53. The "integer part" -2 is not in the right interval. So, one has to use
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
by 5 for getting -2= 3-1\cdot 5, giving :-\frac 13=3-5+\frac 53 = 3-\frac 3, and :\frac 13= 2+3\cdot 5 + \frac 3\cdot 5^2. Similarly, one has :-\frac 23=1-\frac 53, and :\frac 13=2+3\cdot 5 + 1\cdot 5^2 +\frac 3\cdot 5^3. As the "remainder" -\tfrac 13 has already been found, the process can be continued easily, giving coefficients 3 for odd powers of five, and 1 for even powers. Or in the standard 5-adic notation :\frac 13= \ldots 1313132_5 with the
ellipsis The ellipsis , , or (as a single glyph The term glyph is used in typography File:metal movable type.jpg, 225px, Movable type being assembled on a composing stick using pieces that are stored in the type case shown below it Typography ...

ellipsis
\ldots on the left hand side.


''p''-adic series

In this article, given a prime number , a ''-adic series'' is a
formal series In mathematics, and specially in algebra, a formal series is an infinite sum that is considered independently from any notion of convergent series, convergence, and can be manipulated with the usual algebraic operations on series (mathematics), ser ...
of the form :\sum_^\infty a_i p^i, where every nonzero a_i is a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
a_i=\tfrac , such that none of n_i and d_i is divisible by . Every rational number may be viewed as a -adic series with a single term, consisting of its factorization of the form p^k\tfrac nd, with and both coprime with . A -adic series is ''normalized'' if each a_i is an integer in the interval ,p-1 So, the -adic expansion of a rational number is a normalized -adic series. The -adic valuation, or -adic order of a nonzero -adic series is the lowest integer such that a_i\ne 0. The order of the zero series is the infinity \infty. Two -adic series are ''equivalent'' if they have the same order , and if for every integer the difference between their partial sums :\sum_^n a_ip^i-\sum_^n b_ip^i=\sum_^n (a_i-b_i)p^i has an order greater than (that is, is a rational number of the form p^k\tfrac ab, with k>n, and and both coprime with ). For every -adic series S, there is a unique normalized series N such that S and N are equivalent. N is the ''normalization'' of S. The proof is similar to the existence proof of the -adic expansion of a rational number. In particular, every rational number can be considered as a -adic series with a single nonzero term, and the normalization of this series is exactly the rational representation of the rational number. In other words, the equivalence of -adic series is an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, and each
equivalence class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
contains exactly one normalized -adic series. The usual operations of series (addition, subtraction, multiplication, division) map -adic series to -adic series, and are compatible with equivalence of -adic series. That is, denoting the equivalence with , if , and are nonzero -adic series such that S\sim T, one has :\begin S\pm U&\sim T\pm U,\\ SU&\sim TU,\\ 1/S&\sim 1/T. \end Moreover, and have the same order, and the same first term.


Positional notation

It is possible to use a
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications ope ...
similar to that which is used to represent numbers in
base Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider) Base (stylized as BASE) is the third largest of Belgium Belgium ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, ...

base
. Let \sum_^\infty a_i p^i be a normalized -adic series, i.e. each a_i is an integer in the interval ,p-1 One can suppose that k\le 0 by setting a_i=0 for 0\le i (if k>0), and adding the resulting zero terms to the series. If k\ge 0, the positional notation consists of writing the a_i consecutively, ordered by decreasing values of , often with appearing on the right as an index: :\ldots a_n \ldots a_1_p So, the computation of the example above shows that :\frac 13= \ldots 1313132_5, and :\frac 3= \ldots 131313200_5. When k<0, a separating dot is added before the digits with negative index, and, if the index is present, it appears just after the separating dot. For example, :\frac 1= \ldots 3131313._52, and :\frac 1= \ldots 1313131._532. If a -adic representation is finite on the left (that is, a_i=0 for large values of ), then it has the value of a nonnegative rational number of the form n p^v, with n,v integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in
base Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider) Base (stylized as BASE) is the third largest of Belgium Belgium ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, ...

base
. For these rational numbers, the two representations are the same.


Definition

There are several equivalent definitions of -adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concept than those introduced in the preceding sections. Other equivalent definitions use completion of a
discrete valuation ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
(see ),
completion of a metric space In mathematical analysis, a metric space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, ch ...
(see ), or
inverse limit In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s (see ). A -adic number can be defined as a ''normalized -adic series''. Since there are other equivalent definitions that are commonly used, one says often that a normalized -adic series ''represents'' a -adic number, instead of saying that ''it is'' a -adic number. One can say also that any -adic series represents a -adic number, since every -adic series is equivalent to a unique normalized -adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of -adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations on -adic numbers, since the series operations are compatible with equivalence of -adic series. With these operations, -adic numbers form a
field (mathematics) In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers do. A ...
called the field of -adic numbers and denoted \Q_p or \mathbf Q_p. There is a unique field homomorphism from the rational numbers into the -adic numbers, which maps a rational number to its -adic expansion. The image of this homomorphism is commonly identified with the field of rational numbers. This allows considering the -adic numbers as an
extension field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the rational numbers, and the rational numbers as a subfield of the -adic numbers. The ''valuation'' of a nonzero -adic number , commonly denoted v_p(x), is the exponent of in the first nonzero term of every -adic series that represents . By convention, v_p(0)=\infty; that is, the valuation of zero, is \infty. This valuation is a discrete valuation. The restriction of this valuation to the rational numbers is the -adic valuation of \Q, that is, the exponent in the factorization of a rational number as \tfrac nd p^v, with both and
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
with .


''p''-adic integers

The -adic integers are the -adic numbers with a nonnegative valuation. Every
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
is a -adic integer (including zero, since 0<\infty). The rational numbers of the form \tfrac nd p^k with coprime with and k\ge 0 are also -adic integers. The -adic integers form a
commutative ring In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
, denoted \Z_p or \mathbf Z_p that has the following properties. * It is an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, 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 Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
(invertible elements) of \Z_p are the -adic numbers of valuation zero. * It is a
principal ideal domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, such that each
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
is generated by a power of . * It is a
local ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
of
Krull dimension In commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with ...
one, since its only
prime ideal In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
s are the
zero ideal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and the ideal generated by , the unique maximal ideal. * It is a
discrete valuation ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
, since this results from the preceding properties. * It is the completion of the local ring \Z_=\, which is the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
of \Z at the prime ideal p\Z. 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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
of the completion of the localization of the integers at the prime ideal generated by .


Topological properties

The -adic valuation allows defining an
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
on -adic numbers: the -adic absolute value of a nonzero -adic number is :, x, _p = p^, where v_p(x) is the -adic valuation of . The -adic absolute value of 0 is , 0, _p = 0. This is an absolute value that satisfies the strong triangle inequality since, for every and one has * , x, _p = 0 if and only if x=0; * , x, _p\cdot , y, _p = , xy, _p *, x+y, _p\le \max(, x, _p,, y, _p) \le , x, _p + , y, _p. Moreover, if , x, _p \ne , y, _p, one has , x+y, _p = \max(, x, _p,, y, _p). This makes the -adic numbers a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, and even an
ultrametric space In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems f ...
, with the -adic distance defined by d_p(x,y)=, x-y, _p. 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. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every
Cauchy sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the
partial sum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s of a -adic series, and thus a unique normalized -adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized -adic series instead of equivalence classes of Cauchy sequences). As the metric is defined from a discrete valuation, every
open ball In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is also closed. More precisely, the open ball B_r(x) =\ equals the closed ball B_ =\, where is the least integer such that p^< r. Similarly, B_r = B_(x), where is the greatest integer such that p^>r. This implies that the -adic numbers form a
locally compact space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
, and the -adic integers—that is, the ball B_1 B_p(0)—form a
compact space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.


Modular properties

The
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
\Z_p/p^n\Z_p may be identified with the ring \Z/p^n\Z of the integers modulo p^n. This can be shown by remarking that every -adic integer, represented by its normalized -adic series, is congruent modulo p^n with its
partial sum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
\sum_^a_ip^i, whose value is an integer in the interval ,p-1 A straightforward verification shows that this defines a
ring isomorphism In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
from \Z_p/p^n\Z_p to \Z/p^n\Z. The
inverse limit In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of the
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
\Z_p/p^n\Z_p is defined as the ring formed by the sequences a_0, a_1, \ldots such that a_i\in \Z/p^i \Z, and a_i \equiv a_ \pmod , for every . The mapping that maps a normalized -adic series to the sequence of its partial sums is a ring isomorphism from \Z_p to the inverse limit of the \Z_p/p^n\Z_p. This provides another way for defining -adic integers (
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 In numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathem ...

Newton's method
, starting from the inverse modulo ; then, each Newton step computes the inverse modulo p^ from the inverse modulo p^n. The same method can be used for computing the -adic
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
of an integer that is a
quadratic residue In number theory, an integer ''q'' is called a quadratic residue modulo operation, modulo ''n'' if it is Congruence relation, congruent to a Square number, perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv ...
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 \Z_p/p^n\Z_p, as soon p^n is larger than twice the given integer. Hensel lifting is a similar method that allows to "lift" the factorization modulo of a polynomial with integer coefficients to a factorization modulo p^n for large values of . This is commonly used by
polynomial factorization In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
algorithms.


Notation

There are several different conventions for writing -adic expansions. So far this article has used a notation for -adic expansions in which
powers Powers may refer to: Arts, entertainment, and media * Powers (comics), ''Powers'' (comics), an American creator-owned comic book series by Brian Michael Bendis and Michael Avon Oeming ** Powers (American TV series), ''Powers'' (American TV seri ...
of  increase from right to left. With this right-to-left notation the 3-adic expansion of , for example, is written as :\dfrac=\dots 121012102_3. When performing arithmetic in this notation, digits are carried to the left. It is also possible to write -adic expansions so that the powers of increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of is :\dfrac=2.01210121\dots_3\mbox\dfrac=20.1210121\dots_3. -adic expansions may be written with other sets of digits instead of . For example, the 3-adic expansion of 1/5 can be written using
balanced ternary Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integer An integer (from the Latin Latin (, or , ) is a classical language A classical language i ...

balanced ternary
digits as :\dfrac=\dots\underline11\underline11\underline11\underline_ . In fact any set of integers which are in distinct residue classes modulo may be used as -adic digits. In number theory,
Teichmüller representatives
Teichmüller representatives
are sometimes used as digits. is a variant of the -adic representation of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s that was proposed in 1979 by
Eric Hehner The given name Eric, Erich, Erikk, Erik, Erick, or Eirik is derived from the Old Norse name ''Eiríkr'' (or ''Eríkr'' in Old East Norse due to monophthongization). The first element, ''ei-'' may be derived from the older Proto-Norse languag ...
and Nigel Horspool for implementing on computers the (exact) arithmetic with these numbers.


Cardinality

Both \Z_p and \Q_p are
uncountable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and have the
cardinality of the continuum In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
. For \Z_p, this results from the -adic representation, which defines a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
of \Z_p on the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
\^\N. For \Q_p this results from its expression as a
countably infinite In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
union of copies of \Z_p: :\Q_p=\bigcup_^\infty \frac 1\Z_p.


Algebraic closure

contains and is a field of
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
. Because can be written as sum of squares, cannot be turned into an
ordered field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. has only a single proper
algebraic extension In abstract algebra, a field extension ''L''/''K'' is called algebraic if every element of ''L'' is algebraic over ''K'', i.e. if every element of ''L'' is a root In vascular plants, the roots are the plant organ, organs of a plant that are ...
: ; in other words, this quadratic extension is already
algebraically closed In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. By contrast, the
algebraic closure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of , denoted \overline, has infinite degree, that is, has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the -adic valuation to \overline, the latter is not (metrically) complete. Its (metric) completion is called or . Here an end is reached, as is algebraically closed. However unlike this field is not locally compact. and are isomorphic as rings, so we may regard as endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
, and does not provide an explicit example of such an isomorphism (that is, it is not
constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has a ...
). If is a finite
Galois extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of , the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
\text \left (\mathbf/ \mathbf_p \right) is solvable. Thus, the Galois group \text \left (\overline/ \mathbf_p \right) is prosolvable.


Multiplicative group

contains the -th
cyclotomic field In number theory, a cyclotomic field is a number field obtained by field extension, adjoining a complex number, complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra ...
() if and only if . For instance, the -th cyclotomic field is a subfield of if and only if , or . In particular, there is no multiplicative - torsion in , if . Also, is the only non-trivial torsion element in . Given a natural number , the index of the multiplicative group of the -th powers of the non-zero elements of in \mathbf_p^ is finite. The number , defined as the sum of reciprocals of
factorial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, is not a member of any -adic field; but . For one must take at least the fourth power. (Thus a number with similar properties as — namely a -th root of — is a member of \overline for all .)


Local–global principle

Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmet ...

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 In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
it can be solved over the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and over the -adic numbers for every prime . This principle holds, for example, for equations given by
quadratic form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s, but fails for higher polynomials in several indeterminates.


Rational arithmetic with Hensel lifting


Generalizations and related concepts

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 field In mathematics, an algebraic number field (or simply number field) K is a finite Degree of a field extension, degree (and hence algebraic extension, algebraic) field extension of the field (mathematics), field of rational numbers Thus K is a fiel ...
s, in an analogous way. This will be described now. Suppose ''D'' is a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathema ...
and ''E'' is its
field of fractions In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
. Pick a non-zero
prime ideal In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
''P'' of ''D''. If ''x'' is a non-zero element of ''E'', then ''xD'' is a
fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domai ...
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^. 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 fieldIn mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field (mathematics), field. Fre ...
''D''/''P'' is finite, to take for ''c'' the size of ''D''/''P''. For example, when ''E'' is a
number field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
,
Ostrowski's theorem In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value of the absolute value function for real numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the ...
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 fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
), which are seen as encoding "local" information. This is accomplished by
adele ring In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the Complete metric space, com ...
s and idele groups. ''p''-adic integers can be extended to ''p''-adic solenoids \mathbb_p. There's a map from \mathbb_p to the
circle group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
whose fibers are the ''p''-adic integers \mathbb_p, in analogy to how there's a map from \mathbb to the circle whose fibers are \mathbb.


See also

* 1 + 2 + 4 + 8 + ... * ''k''-adic notation *
C-minimal theoryIn model theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis ...
*
Hensel's lemmaIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Locally compact fieldIn algebra, a locally compact field is a topological field whose topology forms a locally compact space. (in particular, it is a Hausdorff space). These kinds of fields were originally introduced in p-adic analysis since the fields \mathbb_p are loca ...
* Mahler's theorem * ''p''-adic quantum mechanics *
Profinite integer In mathematics, a profinite integer is an element of the ring (mathematics), ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p where :\varprojlim \mathbb/n\mathbb indicates the profinite c ...
*
Volkenborn integral In mathematics, in the field of p-adic analysis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which ...


Footnotes


Notes


Citations


References

* *. — Translation into English by
John Stillwell John Colin Stillwell (born 1942) is an Australia Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasman ...

John Stillwell
of ''Theorie der algebraischen Functionen einer Veränderlichen'' (1882). * * * * * * * *


Further reading

* * * * *


External links

*
''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
Efficient p-adic arithmetic
(slides)
Introduction to p-adic numbers
* {{DEFAULTSORT:P-Adic Number Field (mathematics) Number theory