P-adic number

File:3-adic integers with dual colorings.svg

The 3-adic integers, with selected corresponding characters on their Pontryagin dual group

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

For example, comparing the expansion of the rational number $\frac{1}{5}$ in [[Ternary numeral system|base Template:Math]] vs. the Template:Math-adic expansion, $$\begin{array}{rlrl}\frac{1}{5}& =0.01210121\dots (\text{base }3)& & =0\cdot 3^0+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\ \frac{1}{5}& =\dots 121012102(\text{3-adic})& & =\cdots +2\cdot 3^3+1\cdot 3^2+0\cdot 3^1+2\cdot 3^0.\end{array}$$

Formally, given a prime number Template:Mvar, a Template:Mvar-adic number can be defined as a series $$s=\sum _{i=k}^{\mathrm{\infty }}{a}_i{p}^i=a_k{p}^k+a_{k+1}{p}^{k+1}+a_{k+2}{p}^{k+2}+\cdots$$ where Template:Mvar is an integer (possibly negative), and each $a_i$ is an integer such that $0\le a_i<p.$ A Template:Mvar-adic integer is a Template:Mvar-adic number such that $k\ge 0.$

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

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

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

Motivation

Roughly speaking, modular arithmetic modulo a positive integer Template:Mvar consists of "approximating" every integer by the remainder of its division by Template:Mvar, called its residue modulo Template:Mvar. The main property of modular arithmetic is that the residue modulo Template:Mvar 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 Template:Mvar.

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

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

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

Essentially, Template:Mvar-adic numbers allows "taking modulo Template:Math for all Template:Mvar at once". A distinguishing feature of Template:Mvar-adic numbers from ordinary modulo arithmetic is that the set of Template:Mvar-adic numbers ${\mathbb{\mathbb{Q}}}_p$ forms a field, making division by Template:Mvar possible (unlike when working modulo Template:Math). Furthermore, the mapping $\mathbb{\mathbb{Z}}\hookrightarrow {\mathbb{\mathbb{Z}}}_p$ is injective, so not much information is lost when reducing to Template:Mvar-adic numbers.

Informal description

There are multiple ways to understand Template:Mvar-adic numbers.

As a base-p expansion

One way to think about Template:Mvar-adic integers is using "base Template:Mvar". For example, every integer can be written in base Template:Mvar,

$$50=1212_3=1\cdot 3^3+2\cdot 3^2+1\cdot 3^1+2\cdot 3^0$$

Informally, Template:Mvar-adic integers can be thought of as integers in base-Template:Mvar, but the digits extend infinitely to the left.

$$\dots 121012102_3=\cdots +2\cdot 3^3+1\cdot 3^2+0\cdot 3^1+2\cdot 3^0$$

Addition and multiplication on Template:Mvar-adic integers can be carried out similarly to integers in base-Template:Mvar.

When adding together two Template:Mvar-adic integers, for example $\dots 012102_3+\dots 101211_3$, their digits are added with carries being propagated from right to left.

$$\begin{array}{cccccccc}& & & {}_1& {}_1& & {}_1& \\ & \cdots & 0& 1& 2& 1& 0& 2{}_3\\ +& \cdots & 1& 0& 1& 2& 1& 1{}_3\end{array}$$

Multiplication of Template:Mvar-adic integers works similarly via long multiplication. Since addition and multiplication can be performed with Template:Mvar-adic integers, they form a ring, denoted ${\mathbb{\mathbb{Z}}}_p$ or ${\mathbf{𝐙}}_p$.

Note that some rational numbers can also be Template:Mvar-adic integers, even if they aren't integers in a real sense. For example, the rational number Template:Sfrac is a 3-adic integer, and has the 3-adic expansion $\frac{1}{5}=\dots 121012102_3$. However, some rational numbers, such as $\frac{1}{p}$, cannot be written as a Template:Mvar-adic integer. Because of this, Template:Mvar-adic integers are generalized further to Template:Mvar-adic numbers:

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

$$\dots 121012.102_3=\cdots +1\cdot 3^1+2\cdot 3^0+1\cdot 3^{-1}+0\cdot 3^{-2}+2\cdot 3^{-3}$$

Equivalently, every Template:Mvar-adic number is of the form $\frac{x}{p^k}$, where Template:Mvar is a Template:Mvar-adic integer.

For any Template:Mvar-adic number Template:Mvar, its multiplicative inverse $\frac{1}{x}$ is also a Template:Mvar-adic number, which can be computed using a variant of long division. For this reason, the Template:Mvar-adic numbers form a field, denoted ${\mathbb{\mathbb{Q}}}_p$ or ${\mathbf{𝐐}}_p$.

As a sequence of residues mod Template:Mvar^k

Another way to define Template:Mvar-adic integers is by representing it as a sequence of residues $x_e$ mod $p^e$ for each integer $e$,

$$x=(x_1\mathrm{mod}p,x_2\mathrm{mod}{p}^2,x_3\mathrm{mod}{p}^3,\dots )$$

satisfying the compatibility relations $x_i\equiv x_j(\mathrm{mod}{p}^i)$ for $i<j$. In this notation, addition and multiplication of Template:Mvar-adic integers are defined component-wise:

$$x+y=(x_1+y_1\mathrm{mod}p,x_2+y_2\mathrm{mod}{p}^2,x_3+y_3\mathrm{mod}{p}^3,\dots )$$ $$x\cdot y=(x_1\cdot y_1\mathrm{mod}p,x_2\cdot y_2\mathrm{mod}{p}^2,x_3\cdot y_3\mathrm{mod}{p}^3,\dots )$$

This is equivalent to the base-Template:Mvar definition, because the last Template:Mvar digits of a base-Template:Mvar expansion uniquely define its value mod Template:Mvar^k, and vice versa.

This form can also explain why some rational numbers are Template:Mvar-adic integers, even if they are not integers. For example, Template:Sfrac is a 3-adic integer, because its 3-adic expansion consists of the multiplicative inverses of 5 mod 3, 3², 3³, ...

$$\begin{array}{rl}\frac{1}{5}& =(\frac{1}{5}\mathrm{mod}3,\frac{1}{5}\mathrm{mod}{3}^2,\frac{1}{5}\mathrm{mod}{3}^3,\frac{1}{5}\mathrm{mod}{3}^4,\dots )\\ & =(2\mathrm{mod}3,2\mathrm{mod}{3}^2,11\mathrm{mod}{3}^3,65\mathrm{mod}{3}^4,\dots )\end{array}$$

Definition

There are several equivalent definitions of Template:Mvar-adic numbers. The two approaches given below are relatively elementary.

As formal series in base Template:Mvar

A Template:Mvar-adic integer is often defined as a formal power series of the form $$r=\sum _{i=0}^{\mathrm{\infty }}{a}_i{p}^i=a_0+a_{1}p+a_2{p}^2+a_3{p}^3+\cdots$$ where each $a_i\in \{0,1,\dots ,p-1\}$ represents a "digit in base Template:Mvar".

A Template:Mvar-adic unit is a Template:Mvar-adic integer whose first digit is nonzero, i.e. $a_0\ne 0$. The set of all Template:Mvar-adic integers is usually denoted ${\mathbb{\mathbb{Z}}}_p$.

A Template:Mvar-adic number is then defined as a formal Laurent series of the form $$r=\sum _{i=v}^{\mathrm{\infty }}{a}_i{p}^i=a_v{p}^v+a_{v+1}{p}^{v+1}+a_{v+2}{p}^{v+2}+a_{v+3}{p}^{v+3}+\cdots$$ where Template:Mvar is a (possibly negative) integer, and each $a_i\in \{0,1,\dots ,p-1\}$. Equivalently, a Template:Mvar-adic number is anything of the form $\frac{x}{p^k}$, where Template:Mvar is a Template:Mvar-adic integer.

The first index Template:Mvar for which the digit $a_v$ is nonzero in Template:Mvar is called the [[p-adic valuation|Template:Mvar-adic valuation]] of Template:Mvar, denoted $v_p(r)$. If $r=0$, then such an index does not exist, so by convention $v_p(0)=\mathrm{\infty }$.

In this definition, addition, subtraction, multiplication, and division of Template:Mvar-adic numbers are carried out similarly to numbers in base Template:Mvar, with "carries" or "borrows" moving from left to right rather than right to left. As an example in ${\mathbb{\mathbb{Q}}}_3$,

$$\begin{array}{llllllllll}& & & {}_1& & & & {}_1& & {}_1\\ & 2\cdot 3^0& +& 0\cdot 3^1& +& 1\cdot 3^2& +& 2\cdot 3^3& +& 1\cdot 3^4& +\cdots \\ +& 1\cdot 3^0& +& 1\cdot 3^1& +& 2\cdot 3^2& +& 1\cdot 3^3& +& 0\cdot 3^4& +\cdots \end{array}$$

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

With these operations, the set of Template:Mvar-adic numbers form a field, denoted ${\mathbb{\mathbb{Q}}}_p$.

As equivalence classes

The Template:Mvar-adic numbers may also be defined as equivalence classes, in a similar way as the definition of real numbers as equivalence classes of Cauchy sequences. It is fundamentally based on the following lemma:

Every nonzero rational number Template:Mvar can be written $r=p^v\frac{m}{n},$ where Template:Mvar, Template:Mvar, and Template:Mvar are integers and neither Template:Mvar nor Template:Mvar is divisible by Template:Mvar.

The exponent Template:Mvar is uniquely determined by Template:Mvar and is called its Template:Mvar-adic valuation, denoted $v_p(r)$. The proof of the lemma results directly from the fundamental theorem of arithmetic.

A Template:Mvar-adic series is a formal Laurent series of the form $$\sum _{i=v}^{\mathrm{\infty }}{r}_i{p}^i,$$ where $v$ is a (possibly negative) 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 Template:Mvar).

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

Two Template:Mvar-adic series ${\sum }_{i=v}^{\mathrm{\infty }}{r}_i{p}^i$ and ${\sum }_{i=w}^{\mathrm{\infty }}{s}_i{p}^i$ are equivalent if there is an integer Template:Mvar 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 Template:Mvar-adic valuation greater than Template:Mvar.

A Template:Mvar-adic series ${\sum }_{i=v}^{\mathrm{\infty }}{a}_i{p}^i$ is normalized if either all $a_i$ are integers such that $0\le a_i<p,$ and $a_v>0,$ or all $a_i$ are zero. In the latter case, the series is called the zero series.

Every Template:Mvar-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|§ Normalization of a Template:Mvar-adic series]], below.

In other words, the equivalence of Template:Mvar-adic series is an equivalence relation, and each equivalence class contains exactly one normalized Template:Mvar-adic series.

The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of Template:Mvar-adic series. That is, denoting the equivalence with Template:Math, if Template:Mvar, Template:Mvar and Template:Mvar are nonzero Template:Mvar-adic series such that $S\sim T,$ one has $$\begin{array}{rl}S\pm U& \sim T\pm U,\\ SU& \sim TU,\\ 1/S& \sim 1/T.\end{array}$$

With this, the Template:Mvar-adic numbers are defined as the equivalence classes of Template:Mvar-adic series.

The uniqueness property of normalization, allows uniquely representing any Template:Mvar-adic number by the corresponding normalized Template:Mvar-adic series. The compatibility of the series equivalence leads almost immediately to basic properties of Template:Mvar-adic numbers:

• Addition, multiplication and multiplicative inverse of Template:Mvar-adic numbers are defined as for formal power series, followed by the normalization of the result.
• With these operations, the Template:Mvar-adic numbers form a field, which is an extension field of the rational numbers.
• The valuation of a nonzero Template:Mvar-adic number Template:Mvar, commonly denoted $v_p(x)$ is the exponent of Template:Mvar in the first non zero term of the corresponding normalized series; the valuation of zero is $v_p(0)=+\mathrm{\infty }$
• The Template:Mvar-adic absolute value of a nonzero Template:Mvar-adic number Template:Mvar, is $|x{|}_p=p^{-v(x)};$ for the zero Template:Mvar-adic number, one has $|0{|}_p=0.$

Normalization of a p-adic series

Starting with the series ${\sum }_{i=v}^{\mathrm{\infty }}{r}_i{p}^i,$ we wish to arrive at an equivalent series such that the Template:Mvar-adic valuation of $r_v$ is zero. For that, one considers the first nonzero $r_i.$ If its Template:Mvar-adic valuation is zero, it suffices to change Template:Mvar into Template:Mvar, that is to start the summation from Template:Mvar. Otherwise, the Template:Mvar-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 Template:Math 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 $[0,p-1].$ Using Bézout's lemma, write this as $r_i=a_i+p{s}_i$, where $a_i\in [0,p-1]$ and $s_i$ has nonnegative valuation. Then, one gets an 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 Template:Math-adic series.

Other equivalent definitions

Other equivalent definitions use completion of a discrete valuation ring (see Template:Slink), completion of a metric space (see Template:Slink), or inverse limits (see Template:Slink).

A Template:Mvar-adic number can be defined as a normalized Template:Mvar-adic series. Since there are other equivalent definitions that are commonly used, one says often that a normalized Template:Mvar-adic series represents a Template:Mvar-adic number, instead of saying that it is a Template:Mvar-adic number.

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

With these operations, Template:Mvar-adic numbers form a field called the field of Template:Math-adic numbers and denoted ${\mathbb{\mathbb{Q}}}_p$ or ${\mathbf{𝐐}}_p.$ There is a unique field homomorphism from the rational numbers into the Template:Mvar-adic numbers, which maps a rational number to its Template:Mvar-adic expansion. The image of this homomorphism is commonly identified with the field of rational numbers. This allows considering the Template:Math-adic numbers as an extension field of the rational numbers, and the rational numbers as a subfield of the Template:Math-adic numbers.

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

Notation

There are several different conventions for writing Template:Mvar-adic expansions. So far this article has used a notation for Template:Mvar-adic expansions in which powers of Template:Mvar increase from right to left. With this right-to-left notation the 3-adic expansion of $\frac{1}{5},$ for example, is written as $$\frac{1}{5}=\dots 121012102_3.$$

When performing arithmetic in this notation, digits are carried to the left. It is also possible to write Template:Mvar-adic expansions so that the powers of Template:Mvar increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of $\frac{1}{5}$ is $$\frac{1}{5}=2.01210121{\dots }_3\text{ or }\frac{1}{15}=20.1210121{\dots }_3.$$

Template:Mvar-adic expansions may be written with other sets of digits instead of Template:Math Template:Math}. For example, the Template:Math-adic expansion of $\frac{1}{5}$ can be written using balanced ternary digits Template:Math}, with Template:Math representing negative one, as $$\frac{1}{5}=\dots \underset{\_}{1}11\underset{\_}{11}11\underset{\_}{11}11{\underset{\_}{1}}_{\text{3}}.$$

In fact any set of Template:Mvar integers which are in distinct residue classes modulo Template:Mvar may be used as Template:Mvar-adic digits. In number theory, Teichmüller representatives are sometimes used as digits.

Template:Vanchor is a variant of the Template:Mvar-adic representation of rational numbers that was proposed in 1979 by Eric Hehner and Nigel Horspool for implementing on computers the (exact) arithmetic with these numbers. It can be used as a compact way to represent rational numbers, which have an infinite periodic sequence of digits. In this notation, a quote mark (') is used to separate the repeating part from the nonrepeating part. $$\frac{1}{5}=1210{}^{\prime }{2}_3$$

p-adic expansion of rational numbers

The decimal expansion of a positive rational number $r$ is its representation as a series $$r=\sum _{i=k}^{\mathrm{\infty }}{a}_{i}10^{-i},$$ where $k$ is an integer and each $a_i$ is also an integer such that $0\le a_i<10.$ This expansion can be computed by long division of the numerator by the denominator, which is itself based on the following theorem: If $r=\frac{n}{d}$ is a rational number such that $0\le r<1,$ there is an integer $a$ such that $0\le a<10,$ and $10r=a+r^{\prime },$ with $0\le r^{\prime }<1.$ The decimal expansion is obtained by repeatedly applying this result to the remainder $r^{\prime }$ which in the iteration assumes the role of the original rational number $r$.

The Template:Mvar-adic expansion of a rational number can be computed similarly, but with a different division step. Suppose that $r=\frac{n}{d}$ is a rational number with nonnegative valuation (that is, Template:Mvar is not divisible by Template:Mvar). The division step consists of writing $$r=a+p{r}^{\prime }$$ where $a$ is an integer such that $0\le a<p,$ and $r^{\prime }$ has nonnegative valuation.

The integer Template:Mvar can be computed as a modular multiplicative inverse: $a=n{d}^{-1}\mathrm{mod}p$. Because of this, writing Template:Mvar in this way is always possible, and such a representation is unique.

The Template:Mvar-adic expansion of a rational number is eventually periodic. Conversely, a series ${\sum }_{i=k}^{\mathrm{\infty }}{a}_i{p}^i,$ with $0\le a_i<p$ converges (for the Template:Mvar-adic absolute value) to a rational number if and only if it is eventually periodic; in this case, the series is the Template:Mvar-adic expansion of that rational number. The proof is similar to that of the similar result for repeating decimals.

Example

Let us compute the 5-adic expansion of $\frac{1}{3}.$ We can write this number as $\frac{1}{3}=2+5\cdot \frac{-1}{3}$. Thus we use $a=2$ for the first step. $$\frac{1}{3}=2+5^1\cdot \left(\frac{-1}{3}\right)$$ For the next step, we can write the "remainder" $\frac{-1}{3}$ as $\frac{-1}{3}=3+5\cdot \frac{-2}{3}$. Thus we use $a=3$. $$\frac{1}{3}=2+3\cdot 5^1+5^2\cdot \left(\frac{-2}{3}\right)$$ We can write the "remainder" $\frac{-2}{3}$ as $\frac{-2}{3}=1+5\cdot \frac{-1}{3}$. Thus we use $a=1$. $$\frac{1}{3}=2+3\cdot 5^1+1\cdot 5^2+5^3\cdot \left(\frac{-1}{3}\right)$$ Notice that we obtain the "remainder" $\frac{-1}{3}$ again, which means the digits can only repeat from this point on. $$\frac{1}{3}=2+3\cdot 5^1+1\cdot 5^2+3\cdot 5^3+1\cdot 5^4+3\cdot 5^5+1\cdot 5^6+\cdots$$ In the standard 5-adic notation, we can write this as $$\frac{1}{3}=\dots 1313132_5$$ with the ellipsis $\dots$ on the left hand side.

p-adic integers

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

A $p$-adic integer can be represented as a sequence $$x=(x_1\mathrm{mod}p,x_2\mathrm{mod}{p}^2,x_3\mathrm{mod}{p}^3,\dots )$$ of residues $x_e$ mod $p^e$ for each integer $e$, satisfying the compatibility relations $x_i\equiv x_j(\mathrm{mod}{p}^i)$ for $i<j$.

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

The Template:Mvar-adic integers form a commutative ring, denoted ${\mathbb{\mathbb{Z}}}_p$ or ${\mathbf{𝐙}}_p$, that has the following properties.

• It is an integral domain, since it is a subring of a field, or since the first term of the series representation of the product of two non zero Template:Mvar-adic series is the product of their first terms.
• The units (invertible elements) of ${\mathbb{\mathbb{Z}}}_p$ are the Template:Mvar-adic numbers of valuation zero.
• It is a principal ideal domain, such that each ideal is generated by a power of Template:Mvar.
• It is a local ring of Krull dimension one, since its only prime ideals are the zero ideal and the ideal generated by Template:Mvar, the unique maximal ideal.
• It is a discrete valuation ring, since this results from the preceding properties.
• It is the completion of the local ring ${\mathbb{\mathbb{Z}}}_{(p)}=\{\frac{n}{d}|n,d\in \mathbb{\mathbb{Z}},d\in ̸p\mathbb{\mathbb{Z}}\},$ which is the localization of $\mathbb{\mathbb{Z}}$ at the prime ideal $p\mathbb{\mathbb{Z}}.$

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

Topological properties

File:3-adic integers.png

Visual depiction of the 3-adic integers ${\mathbb{\mathbb{Z}}}_3$ as a metric space

The Template:Mvar-adic valuation allows defining an absolute value on Template:Mvar-adic numbers: the Template:Mvar-adic absolute value of a nonzero Template:Mvar-adic number Template:Mvar is $$|x{|}_p=p^{-v_p(x)},$$ where $v_p(x)$ is the Template:Mvar-adic valuation of Template:Mvar. The Template:Mvar-adic absolute value of $0$ is $|0{|}_p=0.$ This is an absolute value that satisfies the strong triangle inequality since, for every Template:Mvar and Template:Mvar:

• $|x{|}_p=0$ 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 $|x{|}_p\ne |y{|}_p,$ then $|x+y{|}_p=\max (|x{|}_p,|y{|}_p).$

This makes the Template:Mvar-adic numbers a metric space, and even an ultrametric space, with the Template:Mvar-adic distance defined by $d_p(x,y)=|x-y{|}_p.$

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

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

This implies that the Template:Mvar-adic numbers ${\mathbb{\mathbb{Q}}}_p$ form a locally compact space (locally compact field), and the Template:Mvar-adic integers ${\mathbb{\mathbb{Z}}}_p$—that is, the ball $B_1[0]=B_p(0)$—form a compact space.

The space of 2-adic integers ${\mathbb{\mathbb{Z}}}_2$ is homeomorphic to the Cantor set ${𝒞}$. This can be seen by considering the continuous 1-to-1 mapping $\psi :{\mathbb{\mathbb{Z}}}_2\to {𝒞}$ defined by $$\psi :a_0+a_{1}2+a_2{2}^2+a_3{2}^3+\cdots ⟼\frac{2a_0}{3}+\frac{2a_1}{3^2}+\frac{2a_2}{3^3}+\frac{2a_3}{3^4}+\cdots$$ Moreover, for any Template:Mvar, ${\mathbb{\mathbb{Z}}}_p$ is homeomorphic to ${\mathbb{\mathbb{Z}}}_2$, and therefore also homeomorphic to the Cantor set.

The Pontryagin dual of the group of Template:Mvar-adic integers is the [[Prüfer group|Prüfer Template:Mvar-group]] $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$, and the Pontryagin dual of the Prüfer Template:Mvar-group is the group of Template:Mvar-adic integers.

Modular properties

The quotient ring ${\mathbb{\mathbb{Z}}}_p/p^n{\mathbb{\mathbb{Z}}}_p$ may be identified with the ring $\mathbb{\mathbb{Z}}/p^n\mathbb{\mathbb{Z}}$ of the integers modulo $p^n.$ This can be shown by remarking that every Template:Mvar-adic integer, represented by its normalized Template:Mvar-adic series, is congruent modulo $p^n$ with its partial sum ${\sum }_{i=0}^{n-1}{a}_i{p}^i,$ whose value is an integer in the interval $[0,p^n-1].$ A straightforward verification shows that this defines a ring isomorphism from ${\mathbb{\mathbb{Z}}}_p/p^n{\mathbb{\mathbb{Z}}}_p$ to $\mathbb{\mathbb{Z}}/p^n\mathbb{\mathbb{Z}}.$

The inverse limit of the rings ${\mathbb{\mathbb{Z}}}_p/p^n{\mathbb{\mathbb{Z}}}_p$ is defined as the ring formed by the sequences $a_0,a_1,\dots$ such that $a_i\in \mathbb{\mathbb{Z}}/p^i\mathbb{\mathbb{Z}}$ and $a_i\equiv a_{i+1}(\mathrm{mod}{p}^i)$ for every Template:Mvar.

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

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

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

The same method can be used for computing the Template:Mvar-adic square root of an integer that is a quadratic residue modulo Template:Mvar. 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 ${\mathbb{\mathbb{Z}}}_p/p^n{\mathbb{\mathbb{Z}}}_p$. 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 is a similar method that allows to "lift" the factorization modulo Template:Mvar of a polynomial with integer coefficients to a factorization modulo $p^n$ for large values of Template:Mvar. This is commonly used by polynomial factorization algorithms.

Cardinality

Both ${\mathbb{\mathbb{Z}}}_p$ and ${\mathbb{\mathbb{Q}}}_p$ are uncountable and have the cardinality of the continuum. For ${\mathbb{\mathbb{Z}}}_p,$ this results from the Template:Mvar-adic representation, which defines a bijection of ${\mathbb{\mathbb{Z}}}_p$ on the power set $\{0,\dots ,p-1{\}}^{\mathbb{\mathbb{N}}}.$ For ${\mathbb{\mathbb{Q}}}_p$ this results from its expression as a countably infinite union of copies of ${\mathbb{\mathbb{Z}}}_p$: $${\mathbb{\mathbb{Q}}}_p=\bigcup _{i=0}^{\mathrm{\infty }}\frac{1}{p^i}{\mathbb{\mathbb{Z}}}_p.$$

Algebraic closure

${\mathbb{\mathbb{Q}}}_p$ contains $\mathbb{\mathbb{Q}}$ and is a field of characteristic Template:Math.

Because Template:Math can be written as sum of squares, ${\mathbb{\mathbb{Q}}}_p$ cannot be turned into an ordered field.

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

Its (metric) completion is denoted ${\mathbb{ℂ}}_p$ or ${\mathrm{\Omega }}_p$, and sometimes called the complex Template:Mvar-adic numbers by analogy to the complex numbers. Here an end is reached, as ${\mathbb{ℂ}}_p$ is algebraically closed. However unlike $\mathbb{ℂ}$ this field is not locally compact.

${\mathbb{ℂ}}_p$ and $\mathbb{ℂ}$ are isomorphic as rings, 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 the axiom of choice, and does not provide an explicit example of such an isomorphism (that is, it is not constructive).

If $K$ is any finite Galois extension of ${\mathbb{\mathbb{Q}}}_p,$ the Galois group $\mathrm{Gal}(K/{\mathbb{\mathbb{Q}}}_p)$ is solvable. Thus, the Galois group $\mathrm{Gal}(\stackrel{‾}{{\mathbb{\mathbb{Q}}}_p}/{\mathbb{\mathbb{Q}}}_p)$ is prosolvable.

Multiplicative group

${\mathbb{\mathbb{Q}}}_p$ contains the Template:Mvar-th cyclotomic field (Template:Math) if and only if Template:Math. For instance, the Template:Mvar-th cyclotomic field is a subfield of ${\mathbb{\mathbb{Q}}}_{13}$ if and only if Template:Math, or Template:Math. In particular, there is no multiplicative Template:Mvar-torsion in ${\mathbb{\mathbb{Q}}}_p$ if Template:Math. Also, Template:Math is the only non-trivial torsion element in ${\mathbb{\mathbb{Q}}}_2$.

Given a natural number Template:Mvar, the index of the multiplicative group of the Template:Mvar-th powers of the non-zero elements of ${\mathbb{\mathbb{Q}}}_p$ in ${\mathbb{\mathbb{Q}}}_p^{\times }$ is finite.

The number Template:Mvar, defined as the sum of reciprocals of factorials, is not a member of any Template:Mvar-adic field; but $e^p\in {\mathbb{\mathbb{Q}}}_p$ for $p\ne 2$. For Template:Math one must take at least the fourth power. (Thus a number with similar properties as Template:Mvar — namely a Template:Mvar-th root of Template:Math — is a member of ${\mathbb{\mathbb{Q}}}_p$ for all Template:Mvar.)

Local–global principle

Helmut Hasse's local–global principle is said to hold for an equation if it can be solved over the rational numbers if and only if it can be solved over the real numbers and over the Template:Mvar-adic numbers for every prime Template:Mvar. This principle holds, for example, for equations given by quadratic forms, but fails for higher polynomials in several indeterminates.

Applications

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

Analysis

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

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

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

Hodge theory

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

Teichmüller theory

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

Quantum physics

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

Generalizations and related concepts

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

Suppose D is a Dedekind domain and E is its field of fractions. Pick a non-zero prime ideal P of D. If x is a non-zero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D. 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 $$|x{|}_P=c^{-{\mathrm{ord}}_P(x)}.$$ Completing with respect to this absolute value Template:Nowrap begin|⋅|_PTemplate:Nowrap end 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 D/P is finite, to take for c the size of D/P.

For example, when E is a number field, Ostrowski's theorem says that every non-trivial non-Archimedean absolute value on E arises as some Template:Nowrap begin|⋅|_PTemplate:Nowrap end. 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), which are seen as encoding "local" information. This is accomplished by adele rings and idele groups.

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

The p-adic integers can also be extended to profinite integers $\widehat{\mathbb{\mathbb{Z}}}$, which can be understood as the direct product of rings $$\widehat{\mathbb{\mathbb{Z}}}=\prod _p{\mathbb{\mathbb{Z}}}_p.$$ Unlike the p-adic integers which only generalize the modulo over prime powers p^k, the profinite integers generalizes the modulo over all natural numbers n.

See also

• Non-Archimedean
• p-adically closed field
• 1 + 2 + 4 + 8 + ⋯
• k-adic notation
• C-minimal theory
• Two's complement

References

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

External links

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