p-adic number

Innumber theory,given aprime number $p$ ,the $p$ -adic numbersform an extension of therational numberswhich is distinct from thereal numbers,though with some similar properties; $p$ -adic numbers can be written in a form similar to (possiblyinfinite)decimals,but with digits based on aprime number $p$ rather than ten, and extending to the left rather than to the right.

For example, comparing the expansion of the rational number ${\tfrac {1}{5}}$ inbase $3$ vs. the $3$ -adic expansion,

{\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.01210121\ldots \ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}=\dots 121012102\ \ ({\text{3-adic}})&&{}=\cdots +2\cdot 3^{3}+1\cdot 3^{2}+0\cdot 3^{1}+2\cdot 3^{0}.\end{alignedat}}

Formally, given a prime number $p$ ,a $p$ -adic number can be defined as aseries

s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots

where $k$ is aninteger(possibly negative), and each $a_{i}$ is an integer such that $0\leq a_{i}<p.$ A $p$ -adic integeris a $p$ -adic number such that $k\geq 0.$

In general the series that represents a $p$ -adic number is notconvergentin the usual sense, but it is convergent for the $p$ -adic absolute value $|s|_{p}=p^{-k},$ where $k$ is the least integer $i$ such that $a_{i}\neq 0$ (if all $a_{i}$ are zero, one has the zero $p$ -adic number, which has $0$ as its $p$ -adic absolute value).

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

$p$ -adic numbers were first described byKurt Henselin 1897,^[1]though, with hindsight, some ofErnst Kummer'searlier work can be interpreted as implicitly using $p$ -adic numbers.^{[note 1]}

Motivation

Roughly speaking,modular arithmeticmodulo a positive integer $n$ consists of "approximating" every integer by the remainder of itsdivisionby $n$ ,called itsresidue modulo $n$ .The main property of modular arithmetic is that the residue modulo $n$ of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo $n$ .If one knows that the absolute value of the result is less than $n/2$ ,this allows a computation of the result which does not involve any integer larger than $n$ .

For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying theChinese remainder theoremfor recovering the result modulo the product of the moduli.

Another method discovered byKurt Henselconsists of using a prime modulus $p$ ,and applyingHensel's lemmafor recovering iteratively the result modulo $p^{2},p^{3},\ldots,p^{n},\ldots$ If the process is continued infinitely, this provides eventually a result which is a $p$ -adic number.

Basic lemmas

The theory of $p$ -adic numbers is fundamentally based on the two following lemmas

Every nonzero rational number can be written ${\textstyle p^{v}{\frac {m}{n}},}$ where $v$ , $m$ ,and $n$ are integers and neither $m$ nor $n$ is divisible by $p$ .The exponent $v$ is uniquely determined by the rational number and is called its $p$ -adic valuation(this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from thefundamental theorem of arithmetic.

Every nonzero rational number $r$ of valuation $v$ can be uniquely written $r=ap^{v}+s,$ where $s$ is a rational number of valuation greater than $v$ ,and $a$ is an integer such that $0<a<p.$

The proof of this lemma results frommodular arithmetic:By the above lemma, ${\textstyle r=p^{v}{\frac {m}{n}},}$ where $m$ and $n$ are integerscoprimewith $p$ .Themodular inverseof $n$ is an integer $q$ such that $nq=1+ph$ for some integer $h$ .Therefore, one has ${\textstyle {\frac {1}{n}}=q-p{\frac {h}{n}},}$ and ${\textstyle r=p^{v}mq-p^{v+1}{\frac {hm}{n}}.}$ TheEuclidean divisionof $mq$ by $p$ gives $mq=pk+a$ where $0<a<p,$ since $mq$ is not divisible by $p$ .So,

r=ap^{v}+p^{v+1}{\frac {kn-hm}{n}},

which is the desired result.

This can be iterated starting from $s$ instead of $r$ ,giving the following.

Given a nonzero rational number $r$ of valuation $v$ and a positive integer $k$ ,there are a rational number $s_{k}$ of nonnegative valuation and $k$ uniquely defined nonnegative integers $a_{0},\ldots,a_{k-1}$ less than $p$ such that $a_{0}>0$ and

r=a_{0}p^{v}+a_{1}p^{v+1}+\cdots +a_{k-1}p^{v+k-1}+p^{v+k}s_{k}.

The $p$ -adic numbers are essentially obtained by continuing this infinitely to produce aninfinite series.

p-adic series

The $p$ -adic numbers are commonly defined by means of $p$ -adic series.

A $p$ -adic seriesis aformal power seriesof the form

\sum _{i=v}^{\infty }r_{i}p^{i},

where $v$ is an integer and the $r_{i}$ are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of $r_{i}$ is not divisible by $p$ ).

Every rational number may be viewed as a $p$ -adic series with a single nonzero term, consisting of its factorization of the form $p^{k}{\tfrac {n}{d}},$ with $n$ and $d$ both coprime with $p$ .

Two $p$ -adic series ${\textstyle \sum _{i=v}^{\infty }r_{i}p^{i}}$ and ${\textstyle \sum _{i=w}^{\infty }s_{i}p^{i}}$ areequivalentif there is an integer $N$ such that, for every integer $n>N,$ the rational number

\sum _{i=v}^{n}r_{i}p^{i}-\sum _{i=w}^{n}s_{i}p^{i}

is zero or has a $p$ -adic valuation greater than $n$ .

A $p$ -adic series ${\textstyle \sum _{i=v}^{\infty }a_{i}p^{i}}$ isnormalizedif either all $a_{i}$ are integers such that $0\leq a_{i}<p,$ and $a_{v}>0,$ or all $a_{i}$ are zero. In the latter case, the series is called thezero series.

Every $p$ -adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see§ Normalization of a $p$ -adic series,below.

In other words, the equivalence of $p$ -adic series is anequivalence relation,and eachequivalence classcontains exactly one normalized $p$ -adic series.

The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of $p$ -adic series. That is, denoting the equivalence with $~$ ,if $S$ , $T$ and $U$ are nonzero $p$ -adic series such that $S\sim T,$ one has

{\begin{aligned}S\pm U&\sim T\pm U,\\SU&\sim TU,\\1/S&\sim 1/T.\end{aligned}}

The $p$ -adic numbers are often defined as the equivalence classes of $p$ -adic series, in a similar way as the definition of the real numbers as equivalence classes ofCauchy sequences.The uniqueness property of normalization, allows uniquely representing any $p$ -adic number by the corresponding normalized $p$ -adic series. The compatibility of the series equivalence leads almost immediately to basic properties of $p$ -adic numbers:

Addition,multiplicationandmultiplicative inverseof $p$ -adic numbers are defined as forformal power series,followed by the normalization of the result.
With these operations, the $p$ -adic numbers form afield,which is anextension fieldof the rational numbers.
Thevaluationof a nonzero $p$ -adic number $x$ ,commonly denoted $v_{p}(x)$ is the exponent of $p$ in the first non zero term of the corresponding normalized series; the valuation of zero is $v_{p}(0)=+\infty$
The $p$ -adic absolute valueof a nonzero $p$ -adic number $x$ ,is $|x|_{p}=p^{-v(x)};$ for the zero $p$ -adic number, one has $|0|_{p}=0.$

Normalization of ap-adic series

Starting with the series ${\textstyle \sum _{i=v}^{\infty }r_{i}p^{i},}$ the first above lemma allows getting an equivalent series such that the $p$ -adic valuation of $r_{v}$ is zero. For that, one considers the first nonzero $r_{i}.$ If its $p$ -adic valuation is zero, it suffices to change $v$ into $i$ ,that is to start the summation from $v$ .Otherwise, the $p$ -adic valuation of $r_{i}$ is $j>0,$ and $r_{i}=p^{j}s_{i}$ where the valuation of $s_{i}$ is zero; so, one gets an equivalent series by changing $r_{i}$ to $0$ and $r_{i+j}$ to $r_{i+j}+s_{i}.$ Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation of $r_{v}$ is zero.

Then, if the series is not normalized, consider the first nonzero $r_{i}$ that is not an integer in the interval $[0,p-1].$ The second above lemma allows writing it $r_{i}=a_{i}+ps_{i};$ one gets n equivalent series by replacing $r_{i}$ with $a_{i},$ and adding $s_{i}$ to $r_{i+1}.$ Iterating this process, possibly infinitely many times, provides eventually the desired normalized $p$ -adic series.

Definition

There are several equivalent definitions of $p$ -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 usecompletionof adiscrete valuation ring(see§ p-adic integers),completion of a metric space(see§ Topological properties), orinverse limits(see§ Modular properties).

A $p$ -adic number can be defined as anormalized $p$ -adic series.Since there are other equivalent definitions that are commonly used, one says often that a normalized $p$ -adic seriesrepresentsa $p$ -adic number, instead of saying that itisa $p$ -adic number.

One can say also that any $p$ -adic series represents a $p$ -adic number, since every $p$ -adic series is equivalent to a unique normalized $p$ -adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of $p$ -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 $p$ -adic numbers, since the series operations are compatible with equivalence of $p$ -adic series.

With these operations, $p$ -adic numbers form afieldcalled thefield of $p$ -adic numbersand denoted $\mathbb {Q} _{p}$ or $\mathbf {Q} _{p}.$ There is a uniquefield homomorphismfrom the rational numbers into the $p$ -adic numbers, which maps a rational number to its $p$ -adic expansion. Theimageof this homomorphism is commonly identified with the field of rational numbers. This allows considering the $p$ -adic numbers as anextension fieldof the rational numbers, and the rational numbers as asubfieldof the $p$ -adic numbers.

Thevaluationof a nonzero $p$ -adic number $x$ ,commonly denoted $v_{p}(x),$ is the exponent of $p$ in the first nonzero term of every $p$ -adic series that represents $x$ .By convention, $v_{p}(0)=\infty;$ that is, the valuation of zero is $\infty.$ This valuation is adiscrete valuation.The restriction of this valuation to the rational numbers is the $p$ -adic valuation of $\mathbb {Q},$ that is, the exponent $v$ in the factorization of a rational number as ${\tfrac {n}{d}}p^{v},$ with both $n$ and $d$ coprimewith $p$ .

p-adic integers

The $p$ -adic integersare the $p$ -adic numbers with a nonnegative valuation.

A $p$ -adic integer can be represented as a sequence

x=(x_{1}\operatorname {mod} p,~x_{2}\operatorname {mod} p^{2},~x_{3}\operatorname {mod} p^{3},~\ldots )

of residues $x e$ mod $p e$ for each integer $e$ ,satisfying the compatibility relations $x_{i}\equiv x_{j}~(\operatorname {mod} p^{i})$ for $i < j$ .

Everyintegeris a $p$ -adic integer (including zero, since $0<\infty$ ). The rational numbers of the form ${\textstyle {\tfrac {n}{d}}p^{k}}$ with $d$ coprime with $p$ and $k\geq 0$ are also $p$ -adic integers (for the reason that $d$ has an inverse mod $p e$ for every $e$ ).

The $p$ -adic integers form acommutative ring,denoted $\mathbb {Z} _{p}$ or $\mathbf {Z} _{p}$ ,that has the following properties.

It is anintegral domain,since it is asubringof a field, or since the first term of the series representation of the product of two non zero $p$ -adic series is the product of their first terms.
Theunits(invertible elements) of $\mathbb {Z} _{p}$ are the $p$ -adic numbers of valuation zero.
It is aprincipal ideal domain,such that eachidealis generated by a power of $p$ .
It is alocal ringofKrull dimensionone, since its onlyprime idealsare thezero idealand the ideal generated by $p$ ,the uniquemaximal ideal.
It is adiscrete valuation ring,since this results from the preceding properties.
It is thecompletionof the local ring $\mathbb {Z} _{(p)}=\{{\tfrac {n}{d}}\mid n,d\in \mathbb {Z},\,d\not \in p\mathbb {Z} \},$ which is thelocalizationof $\mathbb {Z}$ at the prime ideal $p\mathbb {Z}.$

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

Topological properties

The $p$ -adic valuation allows defining anabsolute valueon $p$ -adic numbers: the $p$ -adic absolute value of a nonzero $p$ -adic number $x$ is

|x|_{p}=p^{-v_{p}(x)},

where $v_{p}(x)$ is the $p$ -adic valuation of $x$ .The $p$ -adic absolute value of $0$ is $|0|_{p}=0.$ This is an absolute value that satisfies thestrong triangle inequalitysince, for every $x$ and $y$ one has

$|x|_{p}=0$ if and only if $x=0;$
$|x|_{p}\cdot |y|_{p}=|xy|_{p}$
$|x+y|_{p}\leq \max(|x|_{p},|y|_{p})\leq |x|_{p}+|y|_{p}.$

Moreover, if $|x|_{p}\neq |y|_{p},$ one has $|x+y|_{p}=\max(|x|_{p},|y|_{p}).$

This makes the $p$ -adic numbers ametric space,and even anultrametric space,with the $p$ -adic distance defined by $d_{p}(x,y)=|x-y|_{p}.$

As a metric space, the $p$ -adic numbers form thecompletionof the rational numbers equipped with the $p$ -adic absolute value. This provides another way for defining the $p$ -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 everyCauchy sequencea subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of thepartial sumsof a $p$ -adic series, and thus a unique normalized $p$ -adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized $p$ -adic series instead of equivalence classes of Cauchy sequences).

As the metric is defined from a discrete valuation, everyopen ballis alsoclosed.More precisely, the open ball $B_{r}(x)=\{y\mid d_{p}(x,y)<r\}$ equals the closed ball $B_{p^{-v}}[x]=\{y\mid d_{p}(x,y)\leq p^{-v}\},$ where $v$ is the least integer such that $p^{-v}<r.$ Similarly, $B_{r}[x]=B_{p^{-w}}(x),$ where $w$ is the greatest integer such that $p^{-w}>r.$

This implies that the $p$ -adic numbers form alocally compact space,and the $p$ -adic integers—that is, the ball $B_{1}[0]=B_{p}(0)$ —form acompact space.

p-adic expansion of rational numbers

Thedecimal expansionof a positiverational number $r$ is its representation as aseries

r=\sum _{i=k}^{\infty }a_{i}10^{-i},

where $k$ is an integer and each $a_{i}$ is also anintegersuch that $0\leq a_{i}<10.$ This expansion can be computed bylong divisionof the numerator by the denominator, which is itself based on the following theorem: If $r={\tfrac {n}{d}}$ is a rational number such that $10^{k}\leq r<10^{k+1},$ there is an integer $a$ such that $0<a<10,$ and $r=a\,10^{k}+r',$ with $r'<10^{k}.$ The decimal expansion is obtained by repeatedly applying this result to the remainder $r'$ which in the iteration assumes the role of the original rational number $r$ .

The $p$ -adic expansionof a rational number is defined similarly, but with a different division step. More precisely, given a fixedprime number $p$ ,every nonzero rational number $r$ can be uniquely written as $r=p^{k}{\tfrac {n}{d}},$ where $k$ is a (possibly negative) integer, $n$ and $d$ arecoprime integersboth coprime with $p$ ,and $d$ is positive. The integer $k$ is the $p$ -adic valuationof $r$ ,denoted $v_{p}(r),$ and $p^{-k}$ is its $p$ -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}+r'

where $a$ is an integer such that $0\leq a<p,$ and $r'$ is either zero, or a rational number such that $|r'|_{p}<p^{-k}$ (that is, $v_{p}(r')>k$ ).

The $p$ -adic expansionof $r$ is theformal power series

r=\sum _{i=k}^{\infty }a_{i}p^{i}

obtained by repeating indefinitely theabovedivision step on successive remainders. In a $p$ -adic expansion, all $a_{i}$ are integers such that $0\leq a_{i}<p.$

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 $r$ inbase- $p$ .

The existence and the computation of the $p$ -adic expansion of a rational number results fromBézout's identityin 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 $td+up=1.$ So

r=p^{k}{\tfrac {n}{d}}(td+up)=p^{k}nt+p^{k+1}{\frac {un}{d}}.

Then, theEuclidean divisionof $nt$ by $p$ gives

nt=qp+a,

with $0\leq a<p.$ This gives the division step as

{\begin{array}{lcl}r&=&p^{k}(qp+a)+p^{k+1}{\frac {un}{d}}\\&=&ap^{k}+p^{k+1}\,{\frac {qd+un}{d}},\\\end{array}}

so that in the iteration

r'=p^{k+1}\,{\frac {qd+un}{d}}

is the new rational number.

The uniqueness of the division step and of the whole $p$ -adic expansion is easy: if $p^{k}a_{1}+p^{k+1}s_{1}=p^{k}a_{2}+p^{k+1}s_{2},$ one has $a_{1}-a_{2}=p(s_{2}-s_{1}).$ This means $p$ divides $a_{1}-a_{2}.$ Since $0\leq a_{1}<p$ and $0\leq a_{2}<p,$ the following must be true: $0\leq a_{1}$ and $a_{2}<p.$ 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 $p$ -adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of aconvergent serieswith the $p$ -adic absolute value. In the standard $p$ -adic notation, the digits are written in the same order as in astandard base- $p$ 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 $p$ -adic expansion of a rational number is eventuallyperiodic.Conversely,a series ${\textstyle \sum _{i=k}^{\infty }a_{i}p^{i},}$ with $0\leq a_{i}<p$ converges (for the $p$ -adic absolute value) to a rational numberif and only ifit is eventually periodic; in this case, the series is the $p$ -adic expansion of that rational number. Theproofis similar to that of the similar result forrepeating decimals.

Example

Let us compute the 5-adic expansion of ${\tfrac {1}{3}}.$ 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 theextended Euclidean algorithm). Thus

{\frac {1}{3}}=2+5({\frac {-1}{3}}).

For the next step, one has to expand $-1/3$ (the factor 5 has to be viewed as a "shift"of the $p$ -adic valuation, similar to the basis of any number expansion, and thus it should not be itself expanded). To expand $-1/3$ ,we start from the same Bézout's identity and multiply it by $-1$ ,giving

-{\frac {1}{3}}=-2+{\frac {5}{3}}.

The "integer part" $-2$ is not in the right interval. So, one has to useEuclidean divisionby $5$ for getting $-2=3-1\cdot 5,$ giving

-{\frac {1}{3}}=3-5+{\frac {5}{3}}=3-{\frac {10}{3}}=3+5({\frac {-2}{3}}),

and the expansion in the first step becomes

{\frac {1}{3}}=2+5\cdot (3+5\cdot ({\frac {-2}{3}}))=2+3\cdot 5+{\frac {-2}{3}}\cdot 5^{2}.

Similarly, one has

-{\frac {2}{3}}=1-{\frac {5}{3}},

and

{\frac {1}{3}}=2+3\cdot 5+1\cdot 5^{2}+{\frac {-1}{3}}\cdot 5^{3}.

As the "remainder" $-{\tfrac {1}{3}}$ has already been found, the process can be continued easily, giving coefficients $3$ foroddpowers of five, and $1$ forevenpowers. Or in the standard 5-adic notation

{\frac {1}{3}}=\ldots 1313132_{5}

with theellipsis $\ldots$ on the left hand side.

Positional notation

It is possible to use apositional notationsimilar to that which is used to represent numbers inbase $p$ .

Let ${\textstyle \sum _{i=k}^{\infty }a_{i}p^{i}}$ be a normalized $p$ -adic series, i.e. each $a_{i}$ is an integer in the interval $[0,p-1].$ One can suppose that $k\leq 0$ by setting $a_{i}=0$ for $0\leq i<k$ (if $k>0$ ), and adding the resulting zero terms to the series.

If $k\geq 0,$ the positional notation consists of writing the $a_{i}$ consecutively, ordered by decreasing values of $i$ ,often with $p$ appearing on the right as an index:

\ldots a_{n}\ldots a_{1}{a_{0}}_{p}

So, the computation of theexample aboveshows that

{\frac {1}{3}}=\ldots 1313132_{5},

and

{\frac {25}{3}}=\ldots 131313200_{5}.

When $k<0,$ a separating dot is added before the digits with negative index, and, if the index $p$ is present, it appears just after the separating dot. For example,

{\frac {1}{15}}=\ldots 3131313._{5}2,

and

{\frac {1}{75}}=\ldots 1313131._{5}32.

If a $p$ -adic representation is finite on the left (that is, $a_{i}=0$ for large values of $i$ ), then it has the value of a nonnegative rational number of the form $np^{v},$ with $n,v$ integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation inbase $p$ .For these rational numbers, the two representations are the same.

Modular properties

Thequotient ring $\mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}$ may be identified with thering $\mathbb {Z} /p^{n}\mathbb {Z}$ of the integersmodulo $p^{n}.$ This can be shown by remarking that every $p$ -adic integer, represented by its normalized $p$ -adic series, is congruent modulo $p^{n}$ with itspartial sum ${\textstyle \sum _{i=0}^{n-1}a_{i}p^{i},}$ whose value is an integer in the interval $[0,p^{n}-1].$ A straightforward verification shows that this defines aring isomorphismfrom $\mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}$ to $\mathbb {Z} /p^{n}\mathbb {Z}.$

Theinverse limitof the rings $\mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}$ is defined as the ring formed by the sequences $a_{0},a_{1},\ldots$ such that $a_{i}\in \mathbb {Z} /p^{i}\mathbb {Z}$ and ${\textstyle a_{i}\equiv a_{i+1}{\pmod {p^{i}}}}$ for every $i$ .

The mapping that maps a normalized $p$ -adic series to the sequence of its partial sums is a ring isomorphism from $\mathbb {Z} _{p}$ to the inverse limit of the $\mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}.$ This provides another way for defining $p$ -adic integers (up toan isomorphism).

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

For example, for computing the $p$ -adic (multiplicative) inverse of an integer, one can useNewton's method,starting from the inverse modulo $p$ ;then, each Newton step computes the inverse modulo ${\textstyle p^{n^{2}}}$ from the inverse modulo ${\textstyle p^{n}.}$

The same method can be used for computing the $p$ -adicsquare rootof an integer that is aquadratic residuemodulo $p$ .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 $\mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}$ .Applying Newton's method to find the square root requires ${\textstyle p^{n}}$ to be larger than twice the given integer, which is quickly satisfied.

Hensel liftingis a similar method that allows to "lift" the factorization modulo $p$ of a polynomial with integer coefficients to a factorization modulo ${\textstyle p^{n}}$ for large values of $n$ .This is commonly used bypolynomial factorizationalgorithms.

Notation

There are several different conventions for writing $p$ -adic expansions. So far this article has used a notation for $p$ -adic expansions in whichpowersof $p$ increase from right to left. With this right-to-left notation the 3-adic expansion of ${\tfrac {1}{5}},$ for example, is written as

{\frac {1}{5}}=\dots 121012102_{3}.

When performing arithmetic in this notation, digits arecarriedto the left. It is also possible to write $p$ -adic expansions so that the powers of $p$ increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of ${\tfrac {1}{5}}$ is

{\frac {1}{5}}=2.01210121\dots _{3}{\mbox{ or }}{\frac {1}{15}}=20.1210121\dots _{3}.

$p$ -adic expansions may be written withother sets of digitsinstead of ${0, 1, ...,$ $p - 1$ }. For example, the $3$ -adic expansion of ${\tfrac {1}{5}}$ can be written usingbalanced ternarydigits ${1, 0, 1$ }, with $1$ representing negative one, as

{\frac {1}{5}}=\dots {\underline {1}}11{\underline {11}}11{\underline {11}}11{\underline {1}}_{\text{3}}.

In fact any set of $p$ integers which are in distinctresidue classesmodulo $p$ may be used as $p$ -adic digits. In number theory,Teichmüller representativesare sometimes used as digits.^[2]

Quote notationis a variant of the $p$ -adic representation ofrational numbersthat was proposed in 1979 byEric HehnerandNigel Horspoolfor implementing on computers the (exact) arithmetic with these numbers.^[3]

Cardinality

Both $\mathbb {Z} _{p}$ and $\mathbb {Q} _{p}$ areuncountableand have thecardinality of the continuum.^[4]For $\mathbb {Z} _{p},$ this results from the $p$ -adic representation, which defines abijectionof $\mathbb {Z} _{p}$ on thepower set $\{0,\ldots,p-1\}^{\mathbb {N} }.$ For $\mathbb {Q} _{p}$ this results from its expression as acountably infinite unionof copies of $\mathbb {Z} _{p}$ :

\mathbb {Q} _{p}=\bigcup _{i=0}^{\infty }{\frac {1}{p^{i}}}\mathbb {Z} _{p}.

Algebraic closure

$\mathbb {Q} _{p}$ contains $\mathbb {Q}$ and is a field ofcharacteristic $0$ .

Because $0$ can be written as sum of squares,^[5] $\mathbb {Q} _{p}$ cannot be turned into anordered field.

The field ofreal numbers $\mathbb {R}$ has only a single properalgebraic extension:thecomplex numbers $\mathbb {C}$ .In other words, thisquadratic extensionis alreadyalgebraically closed.By contrast, thealgebraic closureof $\mathbb {Q} _{p}$ ,denoted ${\overline {\mathbb {Q} _{p}}},$ has infinite degree,^[6]that is, $\mathbb {Q} _{p}$ has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the $p$ -adic valuation to ${\overline {\mathbb {Q} _{p}}},$ the latter is not (metrically) complete.^[7]^[8]Its (metric) completion is called $\mathbb {C} _{p}$ or $\Omega _{p}$ .^[8]^[9]Here an end is reached, as $\mathbb {C} _{p}$ is algebraically closed.^[8]^[10]However unlike $\mathbb {C}$ this field is notlocally compact.^[9]

$\mathbb {C} _{p}$ and $\mathbb {C}$ are isomorphic as rings,^[11]so we may regard $\mathbb {C} _{p}$ as $\mathbb {C}$ endowed with an exotic metric. The proof of existence of such a field isomorphism relies on theaxiom of choice,and does not provide an explicit example of such an isomorphism (that is, it is notconstructive).

If $K$ is any finiteGalois extensionof $\mathbb {Q} _{p},$ ,theGalois group $\operatorname {Gal} \left(K/\mathbb {Q} _{p}\right)$ issolvable.Thus, the Galois group $\operatorname {Gal} \left({\overline {\mathbb {Q} _{p}}}/\mathbb {Q} _{p}\right)$ isprosolvable.

Multiplicative group

$\mathbb {Q} _{p}$ contains the $n$ -thcyclotomic field( $n > 2$ ) if and only if $n | p - 1$ .^[12]For instance, the $n$ -th cyclotomic field is a subfield of $\mathbb {Q} _{13}$ if and only if $n = 1, 2, 3, 4, 6$ ,or $12$ .In particular, there is no multiplicative $p$ -torsionin $\mathbb {Q} _{p}$ if $p > 2$ .Also, $-1$ is the only non-trivial torsion element in $\mathbb {Q} _{2}$ .

Given anatural number $k$ ,theindexof the multiplicative group of the $k$ -th powers of the non-zero elements of $\mathbb {Q} _{p}$ in $\mathbb {Q} _{p}^{\times }$ is finite.

The number $e$ ,defined as the sum ofreciprocalsoffactorials,is not a member of any $p$ -adic field; but $e^{p}\in \mathbb {Q} _{p}$ for $p\neq 2$ .For $p = 2$ one must take at least the fourth power.^[13](Thus a number with similar properties as $e$ — namely a $p$ -th root of $e p$ — is a member of $\mathbb {Q} _{p}$ for all $p$ .)

Local–global principle

Helmut Hasse'slocal–global principleis said to hold for an equation if it can be solved over the rational numbersif and only ifit can be solved over the real numbers and over the $p$ -adic numbers for every prime $p$ .This principle holds, for example, for equations given byquadratic forms,but fails for higher polynomials in several indeterminates.

Rational arithmetic with Hensel lifting

Generalizations and related concepts

The reals and the $p$ -adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance generalalgebraic number fields,in an analogous way. This will be described now.

SupposeDis aDedekind domainandEis itsfield of fractions.Pick a non-zeroprime idealPofD.Ifxis a non-zero element ofE,thenxDis afractional idealand can be uniquely factored as a product of positive and negative powers of non-zero prime ideals ofD.We write ord_P(x) for the exponent ofPin this factorization, and for any choice of numbercgreater than 1 we can set

|x|_{P}=c^{-\!\operatorname {ord} _{P}(x)}.

Completing with respect to this absolute value|⋅|_Pyields a fieldE_P,the proper generalization of the field ofp-adic numbers to this setting. The choice ofcdoes not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when theresidue fieldD/Pis finite, to take forcthe size ofD/P.

For example, whenEis anumber field,Ostrowski's theoremsays that every non-trivialnon-Archimedean absolute valueonEarises as some|⋅|_P.The remaining non-trivial absolute values onEarise from the different embeddings ofEinto the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings ofEinto the fieldsC_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 whenEis a number field (or more generally aglobal field), which are seen as encoding "local" information. This is accomplished byadele ringsandidele groups.

p-adic integers can be extended top-adic solenoids $\mathbb {T} _{p}$ .There is a map from $\mathbb {T} _{p}$ to thecircle groupwhose fibers are thep-adic integers $\mathbb {Z} _{p}$ ,in analogy to how there is a map from $\mathbb {R}$ to the circle whose fibers are $\mathbb {Z}$ .

Footnotes

Notes

^Translator's introduction,page 35:"Indeed, with hindsight it becomes apparent that adiscrete valuationis behind Kummer's concept of ideal numbers. "(Dedekind & Weber 2012,p. 35)

Citations

^(Hensel 1897)
^(Hazewinkel 2009,p. 342)
^(Hehner & Horspool 1979,pp. 124–134)
^(Robert 2000,Chapter 1 Section 1.1)
^According toHensel's lemma $\mathbb {Q} _{2}$ contains a square root of $-7$ ,so that $2^{2}+1^{2}+1^{2}+1^{2}+\left({\sqrt {-7}}\right)^{2}=0,$ and if $p > 2$ then also by Hensel's lemma $\mathbb {Q} _{p}$ contains a square root of $1 - p$ ,thus $(p-1)\times 1^{2}+\left({\sqrt {1-p}}\right)^{2}=0.$
^(Gouvêa 1997,Corollary 5.3.10)
^(Gouvêa 1997,Theorem 5.7.4)
^^a ^b ^c(Cassels 1986,p. 149)
^^a ^b(Koblitz 1980,p. 13)
^(Gouvêa 1997,Proposition 5.7.8)
^Two algebraically closed fields are isomorphic if and only if they have the same characteristic and transcendence degree (see, for example Lang’sAlgebraX §1), and both $\mathbb {C} _{p}$ and $\mathbb {C}$ have characteristic zero and the cardinality of the continuum.
^(Gouvêa 1997,Proposition 3.4.2)
^(Robert 2000,Section 4.1)

References

Cassels, J. W. S.(1986),Local Fields,London Mathematical Society Student Texts, vol. 3,Cambridge University Press,ISBN 0-521-31525-5,Zbl 0595.12006
Dedekind, Richard;Weber, Heinrich(2012),Theory of Algebraic Functions of One Variable,History of mathematics, vol. 39, American Mathematical Society,ISBN 978-0-8218-8330-3.— Translation into English byJohn StillwellofTheorie der algebraischen Functionen einer Veränderlichen(1882).
Gouvêa, F. Q. (March 1994), "A Marvelous Proof",American Mathematical Monthly,101(3): 203–222,doi:10.2307/2975598,JSTOR 2975598
Gouvêa, Fernando Q. (1997),p-adic Numbers: An Introduction(2nd ed.), Springer,ISBN 3-540-62911-4,Zbl 0874.11002
Hazewinkel, M., ed. (2009),Handbook of Algebra,vol. 6, North Holland, p. 342,ISBN 978-0-444-53257-2
Hehner, Eric C. R.;Horspool, R. Nigel (1979),"A new representation of the rational numbers for fast easy arithmetic",SIAM Journal on Computing,8(2): 124–134,CiteSeerX10.1.1.64.7714,doi:10.1137/0208011
Hensel, Kurt(1897),"Über eine neue Begründung der Theorie der algebraischen Zahlen",Jahresbericht der Deutschen Mathematiker-Vereinigung,6(3): 83–88
Kelley, John L.(2008) [1955],General Topology,New York: Ishi Press,ISBN 978-0-923891-55-8
Koblitz, Neal(1980),p-adic analysis: a short course on recent work,London Mathematical Society Lecture Note Series, vol. 46,Cambridge University Press,ISBN 0-521-28060-5,Zbl 0439.12011
Robert, Alain M. (2000),A Course inp-adic Analysis,Springer,ISBN 0-387-98669-3

External links

[2] Translator's introduction,page 35:"Indeed, with hindsight it becomes apparent that adiscrete valuationis behind Kummer's concept of ideal numbers. "(Dedekind & Weber 2012,p. 35)

[1] (Hensel 1897)

[3] (Hazewinkel 2009,p. 342)

[4] (Hehner & Horspool 1979,pp. 124–134)

[5] (Robert 2000,Chapter 1 Section 1.1)

[6] According toHensel's lemma $\mathbb {Q} _{2}$ contains a square root of $-7$ ,so that $2^{2}+1^{2}+1^{2}+1^{2}+\left({\sqrt {-7}}\right)^{2}=0,$ and if $p > 2$ then also by Hensel's lemma $\mathbb {Q} _{p}$ contains a square root of $1 - p$ ,thus $(p-1)\times 1^{2}+\left({\sqrt {1-p}}\right)^{2}=0.$

[7] (Gouvêa 1997,Corollary 5.3.10)

[8] (Gouvêa 1997,Theorem 5.7.4)

[C149-9] (Cassels 1986,p. 149)

[K13-10] (Koblitz 1980,p. 13)

[11] (Gouvêa 1997,Proposition 5.7.8)

[12] Two algebraically closed fields are isomorphic if and only if they have the same characteristic and transcendence degree (see, for example Lang’sAlgebraX §1), and both $\mathbb {C} _{p}$ and $\mathbb {C}$ have characteristic zero and the cardinality of the continuum.

[13] (Gouvêa 1997,Proposition 3.4.2)

[14] (Robert 2000,Section 4.1)

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