# P-adic number

The ** p-adic number systems** were first described by Kurt Hensel in 1897. For each prime number

*p*, the

*p*-adic number system extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The main use of these other systems is in number theory.

The extension is achieved by an alternative interpretation of the concept of absolute value. The *p*-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of *p*-adic analysis essentially provides an alternative form of calculus.

More formally, for a given prime *p*, the field **Q**_{p} of *p*-adic numbers is an extension field of the rational numbers. If all of the fields **Q**_{p} are collectively considered, we arrive at Helmut Hasse's local-global principle, which roughly states that certain equations can be solved over the rational numbers if and only if they can be solved over the real numbers *and* over the *p*-adic numbers for every prime *p*. The field **Q**_{p} is also given a topology derived from a metric, which is itself derived from an alternative valuation on the rational numbers. This metric is complete in the sense that every Cauchy sequence converges. This is what allows the development of calculus on **Q**_{p}, and it is the interaction of this analytic and algebraic structure which gives the *p*-adic number systems their power and utility.

In the context of elliptic curves *p*-adic numbers are usually referred to as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell}**
-adic numbers, due to the work of Jean-Pierre Serre. The prime *p* is often reserved for modular arithmetic of such curves.

## Contents

## Motivation

The simplest introduction to *p*-adic numbers is to consider 10-adic
numbers, which are simply integers in which you allow an infinite
number of digits to the left, for example, the number ...9999, and
then do arithmetic with such numbers as usual. In other words, do arithmetic
like you would with real numbers, but with digits going off to the left
instead of to the right. The references to
valuations and metrics given below are simply technical devices which
justify the ordinary operations. For example, one has the computation

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{...9999 \atop +1}} { ...000} }**

which is true because there are an infinite number of carries which never
end, so there will never be a digit "1" on the left in the result. So a first
10-adic result is that ...999 = −1. It follows from this that negative
integers can be represented as digit expansions in which all
lefthand digits are eventually equal to 9. Computer scientists
might be reminded of two's complement notation, in which negative integers
are coded with the leftmost bit being set to 1: in the 2-adic integers, negative integers will correspond to numbers
in which all lefthand digits are eventually equal to 1 (in general, *p* − 1 for *p*-adic
numbers).

One point that confuses many people is why the *p* in *p*-adic numbers is
always prime. As seen above, it is not absolutely necessary, as things work
well enough in base 10. (Often the term *g-adic number* is used when the concept is used for a fixed composite number *g*. for example by Kurt Mahler). However, *p*-adic numbers are most useful for
doing calculus-type computations, and it is important to always be
able to divide, that is, one wants to be able to work in a field. The
point is that *p*-adic numbers form a field if and only if *p* is a
prime power, and you get the same result for a prime power as you do
for the prime (e.g., base 16 is just shorthand for base 2). In particular, if *p* is not a
prime power, then you can always find two
nonzero *p*-adic numbers *A* and *B* such that *AB* = 0, which removes all possibility of finding their inverses. It is an interesting exercise
to find such numbers for *p* = 10, for example, the following (check
that the products are well defined over the 10-adics):

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \prod_{n = 1}^\infty [2 \, (2^{-1} {\rm mod}\, 5^n)], \qquad B = \prod_{n = 1}^\infty [5 \, (5^{-1} {\rm mod}\, 2^n)]. }**

If *p* is a fixed prime number, then any integer can be written as a *p-adic expansion* (writing the number in "base *p*") in the form

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\sum_{i=0}^n a_i p^i}**

where the a_{i} are integers in {0,...,*p* − 1}. This is expressed by saying that the integer has been "written in base *p*". For example, the 2-adic or binary expansion of 35 is 1·2^{5} + 0·2^{4} + 0·2^{3} + 0·2^{2} + 1·2^{1} + 1·2^{0}, often written in the shorthand notation 100011_{2}.

The familiar approach to generalizing this description to the larger domain of the rationals (and, ultimately, to the reals) is to include sums of the form:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\sum_{i=-\infty}^n a_i p^i}**

A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313..._{5}. In this formulation, the integers are precisely those numbers which can be represented in the form where *a*_{i} = 0 for all *i* < 0.

As an alternative, if we extend the *p*-adic expansions by allowing infinite sums of the form

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=k}^{\infty} a_i p^i}**

where *k* is some (not necessarily positive) integer, we obtain the field **Q**_{p} of ** p-adic numbers**. Those

*p*-adic numbers for which

*a*

_{i}= 0 for all

*i*< 0 are also called the

**. The**

*p*-adic integers*p*-adic integers form a subring of

**Q**

_{p}, denoted

**Z**

_{p}. (Note:

**Z**

_{p}is often used to represent the set of integers modulo

*p*. If each set is needed, the latter is usually written

**Z**/

*p*

**Z**or

**Z**/

*p*. Be sure to check the notation for any text you read.)

Intuitively, as opposed to *p*-adic expansions which extend to the *right* as sums of ever smaller, increasingly negative powers of the base *p* (as is done for the real numbers as described above), these are numbers whose *p*-adic expansion to the *left* are allowed to go on forever. For example, the *p*-adic expansion of 1/3 in base 5 is ...1313132, i.e. the limit of the sequence 2, 32, 132, 3132, 13132, 313132, 1313132, ... Informally, we can see that multiplying this "infinite sum" by 3 in base 5 gives ...0000001. As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 is a *p*-adic integer in base 5.

The main technical problem is to define a proper notion of infinite sum which makes these expressions meaningful - this requires the introduction of the *p*-adic metric. Two different but equivalent solutions to this problem are presented below.

## Constructions

### Analytic approach

The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; this allows us to, for example, write 1 as 1.000... = 0.999... . However, the definition of a Cauchy sequence relies on the metric chosen and, by choosing a different one, numbers other than the real numbers can be constructed. The usual metric which yields the real numbers is called the Euclidean metric.

For a given prime *p*, we define the *p-adic metric* in **Q** as follows:
for any non-zero rational number *x*, there is a unique integer *n* allowing us to write *x* = *p*^{n}(*a*/*b*), where neither of the integers *a* and *b* is divisible by *p*. Unless the numerator or denominator of *x* in lowest terms contains *p* as a factor, *n* will be 0. Now define |*x*|_{p} = *p*^{−n}. We also define |0|_{p} = 0.

For example with *x* = 63/550 = 2^{−1} 3^{2} 5^{−2} 7 11^{−1}

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|_2=2}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|_3=1/9}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|_5=25}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|_7=1/7}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|_{11}=11}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|_{\mbox{any other prime}}=1}**

This definition of |*x*|_{p} has the effect that high powers of *p* become "small".

It can be proved that each norm on **Q** is equivalent either to the Euclidean norm or to one of the *p*-adic norms for some prime *p*. The *p*-adic norm defines a metric d_{p} on **Q** by setting

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_p(x,y)=|x-y|_p}**

The field **Q**_{p} of *p*-adic numbers can then be defined as the completion of the metric space (**Q**,d_{p}); its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains **Q**.

It can be shown that in **Q**_{p}, every element *x* may be written in a unique way as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=k}^{\infty} a_i p^i}**

where *k* is some integer and each *a*_{i} is in {0,...,*p* − 1}. This series converges to *x* with respect to the metric d_{p}.

### Algebraic approach

In the algebraic approach, we first define the ring of *p*-adic integers, and then construct the field of quotients of this ring to get the field of *p*-adic numbers.

We start with the inverse limit of the rings
**Z**/*p ^{n}*

**Z**(see modular arithmetic): a

*p*-adic integer is then a sequence (

*a*)

_{n}_{n≥1}such that

*a*is in

_{n}**Z**/

*p*

^{n}**Z**, and if

*n*<

*m*,

*a*≡

_{n}*a*(mod

_{m}*p*).

^{n}Every natural number *m* defines such a sequence (*m* mod *p ^{n}*), and can therefore be regarded as a

*p*-adic integer. For example, in this case 35 as a 2-adic integer would be written as the sequence {1, 3, 3, 3, 3, 35, 35, 35, ...}.

Note that pointwise addition and multiplication of such sequences is well defined, since addition and multiplication commute with the *mod* operator, see modular arithmetic. Also, every sequence (*a _{n}*) where the first element is not 0 has an inverse: since in that case, for every

*n*,

*a*and

_{n}*p*are coprime, and so

*a*and

_{n}*p*are relatively prime. Therefore, each

^{n}*a*has an inverse mod

_{n}*p*, and the sequence of these inverses, (

^{n}*b*), is the sought inverse of (

_{n}*a*).

_{n}Every such sequence can alternatively be written as a series of the form we considered above. For instance, in the 3-adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2*3 + 0*3^{2} + 1*3^{3} + 0*3^{4} + ... The partial sums of this latter series are the elements of the given sequence.

The ring of *p*-adic integers has no zero divisors, so we can take the quotient field to get the field **Q**_{p} of *p*-adic numbers. Note that in this quotient field, every number can be uniquely written as *p ^{−n}u* with a natural number

*n*and a

*p*-adic integer

*u*.

## Properties

The set of *p*-adic integers are the inverse limit of the finite rings **Z**/*p*^{k}**Z**, but are nonetheless uncountable. The endomorphism ring of the Prüfer p-group of rank *n*, denoted **Z**(*p*^{∞})^{n}, is the ring of *n*×*n* matrices over the *p*-adic integers; this is sometimes referred to as the Tate module.

The *p*-adic numbers contain the rational numbers **Q** and form a field of characteristic 0. This field cannot be turned into an ordered field.

The topology of the set of *p*-adic integers is that of a Cantor set; the topology of the set of *p*-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity). In particular, the space of *p*-adic integers is compact while the space of *p*-adic numbers is not; it is only locally compact.
As metric spaces, both the *p*-adic integers and the *p*-adic numbers are complete.

The real numbers have only a single proper algebraic extension, the complex numbers; in other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of the *p*-adic numbers has infinite degree. Furthermore, **Q**_{p} has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, the algebraic closure of **Q**_{p} is not (metrically) complete. Its (metric) completion is called **Ω**_{p}. Here an end is reached, as **Ω**_{p} is algebraically closed.

The field **Ω**_{p} is isomorphic to the field **C** of complex numbers, so we may regard **Ω**_{p} as the complex numbers endowed with an exotic metric. It should be noted that the existence of such a field isomorphism relies on the axiom of choice, and no explicit isomorphism can be given.

The *p*-adic numbers contain the *n*th cyclotomic field if and only if *n* divides *p* − 1. For instance, the *n*th cyclotomic field is a subfield of **Q**_{13} iff *n* = 1, 2, 3, 4, 6, or 12. In particular, there is no *p*-torsion in the *p*-adic numbers.

The number *e*, defined as the sum of reciprocals of factorials, is not a member of any *p*-adic field; but *e ^{p}* is a

*p*-adic number for all

*p*except 2, for which one must take at least the fourth power. Thus

*e*is a member of the algebraic closure of

*p*-adic numbers for all

*p*.

Over the reals, the only functions whose derivative is zero are the constant functions. This is not true over **Q**_{p}. For instance, the function

*f*:**Q**_{p}→**Q**_{p},*f*(*x*) = (1/|*x*|_{p})^{2}for*x*≠ 0,*f*(0) = 0,

has zero derivative everywhere but is not even locally constant at 0.

Given any elements *r*_{∞}, *r*_{2}, *r*_{3}, *r*_{5}, *r*_{7}, ... where *r*_{p} is in **Q**_{p} (and **Q**_{∞} stands for **R**), it is possible to find a sequence (*x*_{n}) in **Q** such that for all *p* (including ∞), the limit of *x*_{n} in **Q**_{p} is *r*_{p}.

The reals and the *p*-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 quotient field. The non-zero prime ideals of *D* are also called *finite places* or *finite primes* of *E*. 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 finite primes of *E*. If *P* is such a finite prime, we write ord_{P}(*x*) for the exponent of *P* in this factorization, and define

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|_P = (NP)^{-\operatorname{ord}_P(x)}}**

where *NP* denotes the (finite) cardinality of *D*/*P*. Completing with respect to this norm |.|_{P} then yields a field *E*_{P}, the proper generalization of the field of *p*-adic numbers to this setting.

Often, one needs to simultaneously keep track of all the above mentioned completions, which are seen as encoding "local" information. This is accomplished by adele rings and idele groups.

## See also

ca:Nombre p-àdic de:P-adische Zahl es:Número p-ádico fr:Nombre p-adique nl:P-adisch getal ja:P進数 ru:P-адическое число tr:P-sel sayılar zh:P進數