In
mathematics, 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 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 because the only way ...
extends the ordinary
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s in a different way from the extension of the rational
number system
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ca ...
to the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (201 ...
and
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or
absolute value. In particular, two -adic numbers are considered to be close when their difference is
divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
by a high
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may ...
of : the higher the power, the closer they are. This property enables -adic numbers to encode
congruence 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 pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
– including, for example, in
the famous proof of
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have bee ...
by
Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awar ...
.
These numbers were first described by
Kurt Hensel
Kurt Wilhelm Sebastian Hensel (29 December 1861 – 1 June 1941) was a German mathematician born in Königsberg.
Life and career
Hensel was born in Königsberg, East Prussia (today Kaliningrad, Russia), the son of Julia (née von Adelson) and lan ...
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 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
methods into 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 generalizati ...
.
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 grass ...
of -adic numbers is a
completion of the rational numbers. The field is also given a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
derived from a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
, which is itself derived from the
-adic order, an alternative
valuation on the rational numbers. This metric space is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
in the sense that every
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
converges to a point in . This is what allows the development of calculus on , and it is the interaction of this analytic and
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
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphorical expression, a particular word, phrase, ...
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
**The dyadic variation of Democratic peace theory
*Dyadic counterpoint, the voice-against-voice conception of polyp ...
or
triadic.
''p''-adic expansion of rational numbers
The
decimal expansion
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, i ...
of a positive
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
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 ...
:
where
is an integer and each
is also an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
such that
This expansion can be computed by
long division
In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals ( Positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps ...
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
.
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 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 because the only way ...
, every nonzero rational number
can be uniquely written as
where
is a (possibly negative) integer,
and
are
coprime integers both coprime with
, and
is positive. The integer
is the -adic valuation of
, denoted
and
is its -adic absolute value, denoted
(the absolute value is small when the valuation is large). The division step consists of writing
:
where
is an integer such that
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
:
r = \sum_^\infty a_i p^i
obtained by repeating indefinitely the
above 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 n 1 with n > 0, 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 r in base-.
The existence and the computation of the -adic expansion of a rational number results from Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem:
Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they ...
in the following way. If, as above,
r=p^k \tfrac n d, and
d and
p are coprime, there exist integers
t and
u such that
t d+u p=1. So
:
r=p^k \tfrac n d(t d+u p)=p^k n t + p^\fracd.
Then, the
Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
of
n t by
p gives
:
n t=q p+a,
with
0\le a
This gives the division step as
:\begin
r & = & p^k(q p+a) + p^\frac d \\
& = & a p^k +p^\,\frac d, \\
\end
so that in the iteration
:r' = p^\,\frac d
is the new rational number.
The uniqueness of the division step and of the whole -adic expansion is easy: if p^k a_1 + p^s_1=p^k a_2 + p^s_2, one has a_1-a_2=p(s_2-s_1). This means p divides a_1-a_2. Since 0\le a_1 and 0\le a_2 the following must be true: 0\le a_1 and a_2 Thus, one gets -p < a_1-a_2 < p, and since p divides a_1-a_2 it must be that a_1=a_2.
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, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted
:S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k.
The th partial ...
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.
Conversely
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
, a series
\sum_^\infty a_i p^i, with
0\le a_i converges (for the -adic absolute value) to a rational number
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
it is eventually periodic; in this case, the series is the -adic expansion of that rational number. The
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
is similar to that of the similar result for
repeating decimal
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational i ...
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 and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's ...
). 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
Shift may refer to:
Art, entertainment, and media Gaming
* ''Shift'' (series), a 2008 online video game series by Armor Games
* '' Need for Speed: Shift'', a 2009 racing video game
** '' Shift 2: Unleashed'', its 2011 sequel
Literature
* ''Sh ...
" 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 dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
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
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
** Even language, a language spoken by the Evens
* Odd and Even, a solitaire ga ...
powers.
Or in the standard 5-adic notation
:
\frac 13= \ldots 1313132_5
with the
ellipsis
The ellipsis (, also known informally as dot dot dot) is a series of dots that indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning. The plural is ellipses. The term origin ...
\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, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
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 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, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
, and each
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
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 similar to that which is used to represent numbers in
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 . 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 concepts than those introduced in the preceding sections. Other equivalent definitions use
completion of a
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R' ...
(see ),
completion of a metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
(see ), or
inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...
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
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 grass ...
called the field of -adic numbers and denoted
\Q_p or
\mathbf Q_p. There is a unique
field homomorphism
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
Definition of a field
A field is a commutative ri ...
from the rational numbers into the -adic numbers, which maps a rational number to its -adic expansion. The
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of this homomorphism is commonly identified with the field of rational numbers. This allows considering the -adic numbers as an
extension field
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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 mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
with .
''p''-adic integers
The -adic integers are the -adic numbers with a nonnegative valuation.
A -adic integer can be represented as a sequence
:
x = (x_1 \operatorname p, ~ x_2 \operatorname p^2, ~ x_3 \operatorname p^3, ~ \ldots)
of residues mod for each integer , satisfying the compatibility relations
x_i \equiv x_j ~ (\operatorname p^i) for .
Every
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 (for the reason that has an inverse mod for every ).
The -adic integers form a
commutative ring, denoted
\Z_p or
\mathbf Z_p, that has the following properties.
* It is an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, since it is a
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
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, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
(invertible elements) of
\Z_p are the -adic numbers of valuation zero.
* It is a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...
, such that each
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
is generated by a power of .
* It is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
of
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...
one, since its only
prime ideals are the
zero ideal
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive ident ...
and the ideal generated by , the unique
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
.
* It is a
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R' ...
, since this results from the preceding properties.
* It is the
completion of the local ring
\Z_ = \, which is the
localization 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, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
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 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
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 ...
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, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
, 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 ...
, 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, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
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, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
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, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are def ...
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 and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
, and the -adic integers—that is, the ball
B_1 B_p(0)—form a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
.
Modular properties
The
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...
\Z_p/p^n\Z_p may be identified with the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
\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, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
\sum_^a_ip^i, whose value is an integer in the interval
,p^n-1 A straightforward verification shows that this defines a
ring isomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
from
\Z_p/p^n\Z_p to
\Z/p^n\Z.
The
inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...
of the rings
\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 objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
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, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real ...
, 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, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of an integer that is a
quadratic residue
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that:
:x^2\equiv q \pmod.
Otherwise, ''q'' is called a quadratic non ...
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. Applying Newton's method to find the square root requires
p^n to be larger than twice the given integer, which is quickly satisfied.
Hensel lifting In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' t ...
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 and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same d ...
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 and media
* ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming
** ''Powers'' (American TV series), a 2015–2016 series based on the comics
* ''Powers'' (British TV series), a 200 ...
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 integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalanc ...
digits as
:
\dfrac=\dots\underline11\underline11\underline11\underline_ .
In fact any set of integers which are in distinct
residue class
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
es modulo may be used as -adic digits. In number theory,
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
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 ''* ai ...
and
Nigel Horspool for implementing on computers the (exact) arithmetic with these numbers.
Cardinality
Both
\Z_p and
\Q_p are
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
and have the
cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \ma ...
. For
\Z_p, this results from the -adic representation, which defines a
bijection of
\Z_p on the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is ...
\^\N. For
\Q_p this results from its expression as a
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of copies of
\Z_p:
:
\Q_p=\bigcup_^\infty \frac 1\Z_p.
Algebraic closure
contains and is a field of
characteristic .
Because can be written as sum of squares, cannot be turned into an
ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fie ...
.
has only a single proper
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field e ...
: ; in other words, this
quadratic extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
is already
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, becaus ...
. By contrast, the
algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
.[
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, 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 ...
).
If is a finite Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...
of , the Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
\operatorname \left(\mathbf/ \mathbf_p \right) is solvable. Thus, the Galois group \operatorname \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 adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because o ...
() 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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
, the index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
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, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) ...
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 working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory a ...
'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 and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
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 form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
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) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
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 which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
and ''E'' is its field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
. Pick a non-zero prime ideal ''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 dom ...
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 field In 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. Frequently, ''R'' is ...
''D''/''P'' is finite, to take for ''c'' the size of ''D''/''P''.
For example, when ''E'' is a number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
, Ostrowski's theorem
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers \Q is equivalent to either the usual real absolute value or a -adic absolute value.
Definitions
Rais ...
says that every non-trivial non-Archimedean absolute value on ''E'' arises as some , . , ''P''. The remaining non-trivial absolute values on ''E'' arise from the different embeddings of ''E'' into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of ''E'' into the fields C''p'', thus putting the description of all
the non-trivial absolute values of a number field on a common footing.)
Often, one needs to simultaneously keep track of all the above-mentioned completions when ''E'' is a number field (or more generally a global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global function fi ...
), which are seen as encoding "local" information. This is accomplished by adele ring
Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a reco ...
s and idele groups.
''p''-adic integers can be extended to ''p''-adic solenoids \mathbb_p. There is a map from \mathbb_p to the circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
whose fibers are the ''p''-adic integers \mathbb_p, in analogy to how there is a map from \mathbb to the circle whose fibers are \mathbb.
See also
* Non-archimedean
* p-adic quantum mechanics
* p-adic Hodge theory
In mathematics, ''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 Q''p''). The theory has its beginnings ...
* p-adic Teichmuller theory
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
* p-adic analysis
In mathematics, ''p''-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers.
The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of l ...
* 1 + 2 + 4 + 8 + ...
* ''k''-adic notation
* C-minimal theory In model theory, a branch of mathematical logic, a C-minimal theory is a theory that is "minimal" with respect to a ternary relation ''C'' with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important ...
* Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' ...
* Locally compact field In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space.. These kinds of fields were originally introduced in p-adic analysis
In mathematics, ''p''-adic analysis is a branch of number the ...
* Mahler's theorem
In mathematics, Mahler's theorem, introduced by , expresses continuous ''p''-adic functions in terms of polynomials. Over any field of characteristic 0, one has the following result:
Let (\Delta f)(x)=f(x+1)-f(x) be the forward difference oper ...
* Profinite integer In mathematics, a profinite integer is an element of the 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 completion of \mat ...
* Volkenborn integral
Footnotes
Notes
Citations
References
*
*. — Translation into English by John Stillwell
John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University.
Biography
He was born in Melbourne, Australia and lived there until he went to the Massachusetts Ins ...
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