View source for P-adic number

{{Short description|Number system extending the rational numbers}}
{{DISPLAYTITLE:''p''-adic number}}
[[Image:3-adic integers with dual colorings.svg|thumb|The 3-adic integers, with selected corresponding characters on their [[Pontryagin dual]] group]]

In [[number theory]], given a [[prime number]] {{mvar|p}},{{efn-num|In this article, unless otherwise stated, {{mvar|p}} denotes a prime number that is fixed once for all.}} the '''{{mvar|p}}-adic numbers''' form an extension of the [[rational number]]s which is distinct from the [[real number]]s, though with some similar properties; {{mvar|p}}-adic numbers can be written in a form similar to (possibly [[infinity (mathematics)|infinite]]) [[decimal representation|decimal]]s, but with digits based on a [[prime number]] {{mvar|p}} rather than ten, and extending to the left rather than to the right.

For example, comparing the expansion of the rational number <math>\tfrac15</math> in [[Ternary numeral system|base {{math|3}}]] vs. the {{math|3}}-adic expansion,
: <math>\begin{alignat}{3}
\tfrac15 &{}= 0.01210121\ldots \ (\text{base } 3)
         &&{}= 0\cdot 3^0 + 0\cdot 3^{-1} + 1\cdot 3^{-2} + 2\cdot 3^{-3} + \cdots \\[5mu]
\tfrac15 &{}= \dots 121012102 \ \ (\text{3-adic})
         &&{}= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0.
\end{alignat}</math>

Formally, given a prime number {{mvar|p}}, a {{mvar|p}}-adic number can be defined as a [[series (mathematics)|series]]
: <math>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</math>
where {{mvar|k}} is an [[integer]] (possibly negative), and each <math>a_i</math> is an integer such that <math>0\le a_i < p.</math> A '''{{mvar|p}}-adic integer''' is a {{mvar|p}}-adic number such that <math>k\ge 0.</math>

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

Every rational number can be uniquely expressed as the sum of a series as above, with respect to the {{mvar|p}}-adic absolute value. This allows considering rational numbers as special {{mvar|p}}-adic numbers, and alternatively defining the {{mvar|p}}-adic numbers as the [[completion (metric space)|completion]] of the rational numbers for the {{mvar|p}}-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.

{{mvar|p}}-adic numbers were first described by [[Kurt Hensel]] in 1897,<ref>{{Harv|Hensel|1897}}</ref> though, with hindsight, some of [[Ernst Kummer|Ernst Kummer's]] earlier work can be interpreted as implicitly using {{mvar|p}}-adic numbers.<ref group="note">Translator's introduction, [https://books.google.com/books?id=Qxte2mhlEOYC&pg=PA35 page 35]: "Indeed, with hindsight it becomes apparent that a [[discrete valuation]] is behind Kummer's concept of ideal numbers." {{Harv|Dedekind|Weber|2012|p=35}}</ref>

== Motivation ==

Roughly speaking, [[modular arithmetic]] modulo a positive integer {{mvar|n}} consists of "approximating" every integer by the remainder of its [[Euclidean division|division]] by {{mvar|n}}, called its ''residue modulo'' {{mvar|n}}. The main property of modular arithmetic is that the residue modulo {{mvar|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 {{mvar|n}}. If one knows that the absolute value of the result is less than {{mvar|n/2}}, this allows a computation of the result which does not involve any integer larger than {{mvar|n}}.

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

Another method discovered by [[Kurt Hensel]] consists of using a prime modulus {{mvar|p}}, and applying [[Hensel's lemma]] for recovering iteratively the result modulo <math>p^2, p^3, \ldots, p^n, \ldots</math> If the process is continued infinitely, this provides eventually  a result which is a {{mvar|p}}-adic number.

== Basic lemmas ==

The theory of {{mvar|p}}-adic numbers is fundamentally based on the two following lemmas:

''Every nonzero rational number can be written <math display=inline>p^v\frac{m}{n},</math> where {{mvar|v}}, {{mvar|m}}, and {{mvar|n}} are integers and neither {{mvar|m}} nor {{mvar|n}} is divisible by {{mvar|p}}.'' The exponent {{mvar|v}} is uniquely determined by the rational number and is called its ''{{mvar|p}}-adic valuation'' (this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from the [[fundamental theorem of arithmetic]].

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

The proof of this lemma results from [[modular arithmetic]]: By the above lemma, <math display=inline>r=p^v\frac{m}{n},</math> where {{mvar|m}} and {{mvar|n}} are integers [[coprime]] with {{mvar|p}}. 
By [[Bezout lemma]], there exist integers {{mvar|a}} and {{mvar|b}}, with <math>0\leq a < p</math>, such that 
<math> m  =  a n + b p.</math> Setting <math> s = b/n</math> (hence <math>{\rm val}(s) \geq 0</math>), we have
:<math> {m\over n} = a + p {b \over n},\quad {\rm or} \quad r = a p^v  + p^{v + 1} s.</math>

To show the uniqueness of this representation, observe that if <math> r = a' p^v + p^{v + 1} s',</math> with 
<math>0\leq a' < p</math> and <math>{\rm val}(s')\geq 0</math>,
there holds by difference <math>(a -a') + p(s- s') = 0,</math> with <math>|a - a'| < p</math> and <math>{\rm val}(s-s') \geq 0</math>.
Write <math> s-s' = u/v</math>, where {{mvar|v}} is coprime to {{mvar|p}}; then 
<math>(a - a')v + p u = 0</math>, which is possible only if <math>a - a' = 0</math> and <math>u=0</math>.
Hence <math>a = a'</math> and <math> s = s'</math>.   

The above process can be iterated starting from {{mvar|s}} instead of {{mvar|r}}, giving the following.

''Given a nonzero rational number {{mvar|r}} of valuation {{mvar|v}} and a positive integer {{mvar|k}}, there are a rational number <math>s_k</math> of nonnegative valuation and {{mvar|k}} uniquely  defined nonnegative integers <math>a_0, \ldots, a_{k-1}</math> less than {{mvar|p}} such that <math>a_0>0</math> and''
: <math>r=a_0p^v + a_1 p^{v+1} +\cdots + a_{k-1}p^{v+k-1} +p^{v+k}s_k.</math>

The {{mvar|p}}-adic numbers are essentially obtained by continuing this infinitely to produce an [[infinite series]].

== ''p''-adic series ==

The {{mvar|p}}-adic numbers are commonly defined by means of {{mvar|p}}-adic series.

A ''{{mvar|p}}-adic series'' is a [[formal power series]] of the form 
: <math>\sum_{i=v}^\infty r_i p^{i},</math>
where <math>v</math> is an integer and the <math>r_i</math> are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of <math>r_i</math> is not divisible by {{mvar|p}}).

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

Two {{mvar|p}}-adic series <math display=inline>\sum_{i=v}^\infty r_i p^{i} </math> and <math display=inline> \sum_{i=w}^\infty s_i p^{i} </math>
are ''equivalent'' if there is an integer {{mvar|N}} such that, for every integer <math>n>N,</math> the rational number 
: <math>\sum_{i=v}^n r_i p^{i} - \sum_{i=w}^n s_i p^{i} </math> 
is zero or has a {{mvar|p}}-adic valuation greater than {{mvar|n}}.

A {{mvar|p}}-adic series <math display=inline>\sum_{i=v}^\infty a_i p^{i} </math> is ''normalized'' if either all <math>a_i</math> are integers such that <math>0\le a_i <p,</math> and <math>a_v >0,</math> or all <math>a_i</math> are zero. In the latter case, the series is called the ''zero series''.

Every {{mvar|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|§ Normalization of a {{mvar|p}}-adic series]], below.

In other words, the equivalence of {{mvar|p}}-adic series is an [[equivalence relation]], and each [[equivalence class]] contains exactly one normalized {{mvar|p}}-adic series.

The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of {{mvar|p}}-adic series. That is, denoting the equivalence with {{math|~}}, if {{mvar|S}}, {{mvar|T}} and {{mvar|U}} are nonzero {{mvar|p}}-adic series such that <math>S\sim T,</math> one has
: <math>\begin{align}
S\pm U&\sim T\pm U,\\
SU&\sim TU,\\
1/S&\sim 1/T.
\end{align}</math>

The {{mvar|p}}-adic numbers are often defined as the equivalence classes of {{mvar|p}}-adic series, in a similar way as the definition of the real numbers as equivalence classes of [[Cauchy sequence]]s. The uniqueness property of normalization, allows uniquely representing any {{mvar|p}}-adic number by the corresponding normalized {{mvar|p}}-adic series. The compatibility of the series equivalence leads almost immediately to basic properties of {{mvar|p}}-adic numbers:
* ''Addition'', ''multiplication'' and [[multiplicative inverse]] of {{mvar|p}}-adic numbers are defined as for [[formal power series]], followed by the normalization of the result. 
* With these operations, the {{mvar|p}}-adic numbers form a [[field (mathematics)|field]], which is an [[extension field]] of the rational numbers.
* The ''valuation'' of a nonzero {{mvar|p}}-adic number {{mvar|x}}, commonly denoted <math>v_p(x)</math> is the exponent of {{mvar|p}} in the first non zero term of the corresponding normalized series; the valuation of zero is <math>v_p(0)=+\infty</math>
* The ''{{mvar|p}}-adic absolute value'' of a nonzero {{mvar|p}}-adic number {{mvar|x}}, is <math>|x|_p=p^{-v(x)};</math> for the zero {{mvar|p}}-adic number, one has <math>|0|_p=0.</math>

=== Normalization of a ''p''-adic series ===
Starting with the series <math display=inline>\sum_{i=v}^\infty r_i p^{i}, </math> the first above lemma allows getting an equivalent series such that the {{mvar|p}}-adic valuation of <math>r_v</math> is zero. For that, one considers the first nonzero <math>r_i.</math> If its {{mvar|p}}-adic valuation is zero, it suffices to change {{mvar|v}} into {{mvar|i}}, that is to start the summation from {{mvar|v}}. Otherwise, the {{mvar|p}}-adic valuation of <math>r_i</math> is <math>j>0,</math> and <math>r_i= p^js_i</math> where the valuation of <math>s_i</math> is zero; so, one gets an equivalent series by changing <math>r_i</math> to {{math|0}} and <math>r_{i+j}</math> to <math>r_{i+j} + s_i.</math> 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 <math>r_v</math> is zero.

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

== Definition ==
There are several equivalent definitions of {{mvar|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 use [[completion of a ring|completion]] of a [[discrete valuation ring]] (see {{slink||p-adic integers}}), [[completion of a metric space]] (see {{slink||Topological properties}}), or [[inverse limit]]s (see {{slink||Modular properties}}).

A {{mvar|p}}-adic number can be defined as a ''normalized {{mvar|p}}-adic series''. Since there are other equivalent definitions that are commonly used, one says often that a normalized {{mvar|p}}-adic series ''represents'' a {{mvar|p}}-adic number, instead of saying that it ''is'' a {{mvar|p}}-adic number.

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

{{anchor|Field of p-adic numbers}}
With these operations, {{mvar|p}}-adic numbers form a [[field (mathematics)|field]] called the '''field of {{math|''p''}}-adic numbers''' and denoted <math>\Q_p</math> or <math>\mathbf Q_p.</math> There is a unique [[field homomorphism]] from the rational numbers into the {{mvar|p}}-adic numbers, which maps a rational number to its {{mvar|p}}-adic expansion. The [[image (mathematics)|image]] of this homomorphism is commonly identified with the field of rational numbers. This allows considering the {{math|''p''}}-adic numbers as an [[extension field]] of the rational numbers, and the rational numbers as a [[subfield (mathematics)|subfield]] of the {{math|''p''}}-adic numbers.

The ''valuation'' of a nonzero {{mvar|p}}-adic number {{mvar|x}}, commonly denoted <math>v_p(x),</math> is the exponent of {{mvar|p}} in the first nonzero term of every {{mvar|p}}-adic series that represents {{mvar|x}}. By convention, <math>v_p(0)=\infty;</math> that is, the valuation of zero is <math>\infty.</math> This valuation is a [[discrete valuation]]. The restriction of this valuation to the rational numbers is the {{mvar|p}}-adic valuation of <math>\Q,</math> that is, the exponent {{mvar|v}} in the factorization of a rational number as <math dosplay=inline≝>\tfrac nd p^v,</math> with both {{mvar|n}} and {{mvar|d}} [[coprime]] with {{mvar|p}}.

== ''p''-adic integers ==
The '''{{mvar|p}}-adic integers''' are the {{mvar|p}}-adic numbers with a nonnegative valuation.

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

Every [[integer]] is a <math>p</math>-adic integer (including zero, since <math>0<\infty</math>). The rational numbers of the form <math display=inline> \tfrac nd p^k</math> with <math>d</math> coprime with <math>p</math> and <math>k\ge 0</math> are also <math>p</math>-adic integers (for the reason that <math>d</math> has an inverse mod <math>p^e</math> for every <math>e</math>).

The {{mvar|p}}-adic integers form a [[commutative ring]], denoted <math>\Z_p</math>  or <math>\mathbf Z_p</math>, 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 {{mvar|p}}-adic series is the product of their first terms.
* The [[unit (ring theory)|units]] (invertible elements) of <math>\Z_p</math> are the {{mvar|p}}-adic numbers of valuation zero.
* It is a [[principal ideal domain]], such that each [[ideal (ring theory)|ideal]] is generated by a power of {{mvar|p}}.
* It is a [[local ring]] of [[Krull dimension]] one, since its only [[prime ideal]]s are the [[zero ideal]] and the ideal generated by {{mvar|p}}, the unique [[maximal ideal]].
* It is a [[discrete valuation ring]], since this results from the preceding properties.
* It is the [[completion of a ring|completion]] of the local ring <math>\Z_{(p)} = \{\tfrac nd \mid n, d \in \Z,\, d \not\in p\Z \},</math> which is the [[localization (commutative algebra)|localization]] of <math>\Z</math> at the prime ideal <math>p\Z.</math>

The last property provides a definition of the {{mvar|p}}-adic numbers that is equivalent to the above one: the field of the {{mvar|p}}-adic numbers is the [[field of fractions]] of the completion of the localization of the integers at the prime ideal generated by {{mvar|p}}.

== Topological properties ==
The {{mvar|p}}-adic valuation allows defining an [[absolute value (algebra)|absolute value]] on {{mvar|p}}-adic numbers: the {{mvar|p}}-adic absolute value of a nonzero {{mvar|p}}-adic number {{mvar|x}} is
: <math>|x|_p = p^{-v_p(x)},</math>
where <math>v_p(x)</math> is the {{mvar|p}}-adic valuation of {{mvar|x}}. The {{mvar|p}}-adic absolute value of <math>0</math> is <math>|0|_p = 0.</math> This is an absolute value that satisfies the [[strong triangle inequality]] since, for every {{mvar|x}} and {{mvar|y}} one has
* <math>|x|_p = 0</math> if and only if <math>x=0;</math>
* <math>|x|_p\cdot |y|_p = |xy|_p</math>  
* <math>|x+y|_p\le \max(|x|_p,|y|_p) \le |x|_p + |y|_p.</math>

Moreover, if <math>|x|_p \ne |y|_p,</math> one has <math>|x+y|_p = \max(|x|_p,|y|_p).</math>

This makes the {{mvar|p}}-adic numbers a [[metric space]], and even an [[ultrametric space]], with the {{mvar|p}}-adic distance defined by
<math>d_p(x,y)=|x-y|_p.</math>

As a metric space, the {{mvar|p}}-adic numbers form the [[completion (metric space)|completion]] of the rational numbers equipped with the {{mvar|p}}-adic absolute value. This provides another way for defining the {{mvar|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 every [[Cauchy sequence]] 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]]s of a {{mvar|p}}-adic series, and thus a unique normalized {{mvar|p}}-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized {{mvar|p}}-adic series instead of equivalence classes of Cauchy sequences).

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

This implies that the {{mvar|p}}-adic numbers form a [[locally compact space]], and the {{mvar|p}}-adic integers—that is, the ball <math>B_1[0]=B_p(0)</math>—form a [[compact space]].

== ''p''-adic expansion of rational numbers ==

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

The {{mvar|p}}-''adic expansion'' of a rational number is defined similarly, but with a different division step. More precisely, given a fixed [[prime number]] <math>p</math>, every nonzero rational number <math>r</math> can be uniquely written as <math>r=p^k\tfrac n d,</math> where <math>k</math> is a (possibly negative) integer, <math>n</math> and <math>d</math> are [[coprime integer]]s both coprime with <math>p</math>, and <math>d</math> is positive. The integer <math>k</math> is the '''{{mvar|p}}-adic valuation''' of <math>r</math>, denoted <math>v_p(r),</math> and <math>p^{-k}</math> is its '''{{mvar|p}}-adic absolute value''', denoted <math>|r|_p</math> (the absolute value is small when the valuation is large). The division step consists of writing
: {{anchor|division_step}}<math>r = a\,p^k + r'</math>
where <math>a</math> is an integer such that <math>0\le a <p,</math> and <math>r'</math> is either zero, or a rational number such that <math>|r'|_p < p^{-k}</math> (that is, <math>v_p(r')>k</math>).

The <math>p</math>-''adic expansion'' of <math>r</math> is the [[formal power series]] 
: <math>r = \sum_{i=k}^\infty a_i p^i</math>
obtained by repeating indefinitely the [[#division_step|above]] division step on successive remainders. In a {{mvar|p}}-adic expansion, all <math>a_i</math> are integers such that <math>0\le a_i <p.</math>

If <math>r=p^k \tfrac n 1</math> with <math>n > 0</math>, 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 <math>r</math> in [[base-N|base-{{mvar|p}}]].

The existence and the computation of the {{mvar|p}}-adic expansion of a rational number results from [[Bézout's identity]] in the following way. If, as above, <math>r=p^k \tfrac n d,</math> and <math>d</math> and <math>p</math> are coprime, there exist integers <math>t</math> and <math>u</math> such that <math>t d+u p=1.</math> So
: <math>r=p^k \tfrac n d(t d+u p)=p^k n t + p^{k+1}\frac{u n}d.</math>
Then, the [[Euclidean division]] of <math>n t</math> by <math>p</math> gives
: <math>n t=q p+a,</math>
with <math>0\le a <p.</math>
This gives the division step as 
: <math>\begin{array}{lcl}
r & = & p^k(q p+a) + p^{k+1}\frac {u n}d \\
& = & a p^k +p^{k+1}\,\frac{q d+u n} d, \\
\end{array}</math>
so that in the iteration
: <math>r' = p^{k+1}\,\frac{q d+u n} d</math>
is the new rational number.

The uniqueness of the division step and of the whole {{mvar|p}}-adic expansion is easy: if <math>p^k a_1 + p^{k+1}s_1=p^k a_2 + p^{k+1}s_2,</math> one has <math>a_1-a_2=p(s_2-s_1).</math> This means <math>p</math> divides <math>a_1-a_2.</math> Since <math>0\le a_1 <p</math> and <math>0\le a_2 <p,</math> the following must be true: <math>0\le a_1</math> and <math>a_2<p.</math> Thus, one gets <math>-p < a_1-a_2 < p,</math> and since <math>p</math> divides <math>a_1-a_2</math> it must be that <math>a_1=a_2.</math>

The {{mvar|p}}-adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a [[convergent series]] with the {{mvar|p}}-adic absolute value.
In the standard {{mvar|p}}-adic notation, the digits are written in the same order as in a [[Positional notation#Base of the numeral system|standard base-{{mvar|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 {{mvar|p}}-adic expansion of a rational number is eventually [[periodic function|periodic]]. [[Converse (logic)|Conversely]], a series <math display=inline>\sum_{i=k}^\infty a_i p^i,</math> with <math>0\le a_i <p</math> converges (for the {{mvar|p}}-adic absolute value) to a rational number [[if and only if]] it is eventually periodic; in this case, the series is the {{mvar|p}}-adic expansion of that rational number. The [[mathematical proof|proof]] is similar to that of the similar result for [[repeating decimal]]s.

=== Example ===
Let us compute the 5-adic expansion of <math>\tfrac 13.</math> Bézout's identity for 5 and the denominator 3 is <math>2\cdot 3 + (-1)\cdot 5 =1</math> (for larger examples, this can be computed with the [[extended Euclidean algorithm]]). Thus
: <math>\frac 13= 2+5(\frac {-1}3).</math>
For the next step, one has to expand <math>-1/3</math> (the factor 5 has to be viewed as a "[[arithmetic shift|shift]]" of the {{mvar|p}}-adic valuation, similar to the basis of any number expansion, and thus it should not be itself expanded).  To expand <math>-1/3</math>, we start from the same Bézout's identity and multiply it by <math>-1</math>, giving
: <math>-\frac 13=-2+\frac 53.</math>
The "integer part" <math>-2</math> is not in the right interval. So, one has to use [[Euclidean division]] by <math>5</math> for getting <math>-2= 3-1\cdot 5,</math> giving
: <math>-\frac 13=3-5+\frac 53 = 3-\frac {10}3 = 3 +5 (\frac{-2}3),</math>
and the expansion in the first step becomes
: <math>\frac 13= 2+5\cdot (3 + 5 \cdot (\frac{-2}3))= 2+3\cdot 5 + \frac {-2}3\cdot 5^2.</math>

Similarly, one has 
: <math>-\frac 23=1-\frac 53,</math>
and
: <math>\frac 13=2+3\cdot 5 + 1\cdot 5^2 +\frac {-1}3\cdot 5^3.</math>

As the "remainder" <math>-\tfrac 13</math> has already been found, the process can be continued easily, giving coefficients <math>3</math> for [[parity (mathematics)|odd]] powers of five, and <math>1</math> for [[parity (mathematics)|even]] powers.
Or in the standard 5-adic notation
: <math>\frac 13= \ldots 1313132_5 </math>
with the [[ellipsis]] <math> \ldots </math> on the left hand side.

=== Positional notation ===
It is possible to use a [[positional notation]] similar to that which is used to represent numbers in [[radix|base]] {{mvar|p}}.

Let <math display = inline>\sum_{i=k}^\infty a_i p^i</math> be a normalized {{mvar|p}}-adic series, i.e. each <math>a_i</math> is an integer in the interval <math>[0,p-1].</math> One can suppose that <math>k\le 0</math> by setting <math>a_i=0</math> for <math>0\le i <k</math> (if <math>k>0</math>), and adding the resulting zero terms to the series.

If <math>k\ge 0,</math> the positional notation consists of writing the <math>a_i</math> consecutively, ordered by decreasing values of {{mvar|i}}, often with {{mvar|p}} appearing on the right as an index:
: <math>\ldots a_n \ldots a_1{a_0}_p</math>
So, the computation of the [[#Example|example above]] shows that 
: <math>\frac 13= \ldots 1313132_5,</math>
and 
: <math>\frac {25}3= \ldots 131313200_5.</math>
When <math>k<0,</math> a separating dot is added before the digits with negative index, and, if the index {{mvar|p}} is present, it appears just after the separating dot. For example,
: <math>\frac 1{15}= \ldots 3131313._52,</math>
and
: <math>\frac 1{75}= \ldots 1313131._532.</math>

If a {{mvar|p}}-adic representation is finite on the left (that is, <math>a_i=0</math> for large values of {{mvar|i}}), then it has the value of a nonnegative rational number of the form <math>n p^v,</math> with <math>n,v</math> integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in [[radix|base]] {{mvar|p}}. For these rational numbers, the two representations are the same.

== Modular properties ==
The [[quotient ring]] <math>\Z_p/p^n\Z_p</math> may be identified with the [[ring (mathematics)|ring]] <math>\Z/p^n\Z</math> of the integers [[modular arithmetic|modulo]] <math>p^n.</math> This can be shown by remarking that every {{mvar|p}}-adic integer, represented by its normalized {{mvar|p}}-adic series, is congruent modulo <math>p^n</math> with its [[partial sum]] <math display = inline>\sum_{i=0}^{n-1}a_ip^i,</math> whose value is an integer in the interval <math>[0,p^n-1].</math> A straightforward verification shows that this defines a [[ring isomorphism]] from <math>\Z_p/p^n\Z_p</math> to <math>\Z/p^n\Z.</math>

The [[inverse limit]] of the rings <math>\Z_p/p^n\Z_p</math> is defined as the ring formed by the sequences <math>a_0, a_1, \ldots</math> such that <math>a_i \in \Z/p^i \Z</math> and <math display = inline>a_i \equiv a_{i+1} \pmod {p^i}</math> for every {{mvar|i}}.

The mapping that maps a normalized {{mvar|p}}-adic series to the sequence of its partial sums is a ring isomorphism from <math>\Z_p</math> to the inverse limit of the <math>\Z_p/p^n\Z_p.</math> This provides another way for defining {{mvar|p}}-adic integers ([[up to]] an isomorphism).

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

For example, for computing the {{mvar|p}}-adic (multiplicative) inverse of an integer, one can use [[Newton's method]], starting from the inverse modulo {{mvar|p}}; then, each Newton step computes the inverse modulo <math display = inline>p^{n^2}</math> from the inverse modulo <math display = inline>p^n.</math>

The same method can be used for computing the {{mvar|p}}-adic [[square root]] of an integer that is a [[quadratic residue]] modulo {{mvar|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 <math>\Z_p/p^n\Z_p</math>. Applying Newton's method to find the square root requires <math display = inline>p^n</math> 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 {{mvar|p}} of a polynomial with integer coefficients to a factorization modulo <math display = inline>p^n</math> for large values of {{mvar|n}}. This is commonly used by [[polynomial factorization]] algorithms.

== Notation ==
There are several different conventions for writing {{mvar|p}}-adic expansions. So far this article has used a notation for {{mvar|p}}-adic expansions in which [[exponentiation|powers]] of {{mvar|p}} increase from right to left. With this right-to-left notation the 3-adic expansion of <math>\tfrac15,</math> for example, is written as
: <math>\frac15 = \dots 121012102_3.</math>

When performing arithmetic in this notation, digits are [[carry (arithmetic)|carried]] to the left. It is also possible to write {{mvar|p}}-adic expansions so that the powers of {{mvar|p}} increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of <math>\tfrac15</math> is
: <math>
\frac15 = 2.01210121\dots_3 \mbox{ or }
\frac1{15} = 20.1210121\dots_3.
</math>

{{mvar|p}}-adic expansions may be written with [[Signed-digit representation|other sets of digits]] instead of {{math|{0,&thinsp;1,&thinsp;...,}}&thinsp;{{math|''p'' − 1}}}. For example, the {{math|3}}-adic expansion of <math>\tfrac15</math> can be written using [[balanced ternary]] digits {{math|{<u>1</u>,&thinsp;0,&thinsp;1}}}, with {{math|<u>1</u>}} representing negative one, as
: <math>\frac15 = \dots\underline{1}11\underline{11}11\underline{11}11\underline{1}_{\text{3}} .</math>

In fact any set of {{mvar|p}} integers which are in distinct [[residue class]]es modulo {{mvar|p}} may be used as {{mvar|p}}-adic digits. In number theory, [[Witt vector#Motivation|Teichmüller representatives]] are sometimes used as digits.<ref>{{Harv|Hazewinkel|2009|p=342}}</ref>

'''{{vanchor|Quote notation}}''' is a variant of the {{mvar|p}}-adic representation of [[rational number]]s that was proposed in 1979 by [[Eric Hehner]] and [[Nigel Horspool]] for implementing on computers the (exact) arithmetic with these numbers.<ref>{{Harv|Hehner|Horspool|1979|pp=124–134}}</ref>

== Cardinality ==
Both <math>\Z_p</math> and <math>\Q_p</math> are [[uncountable set|uncountable]] and have the [[cardinality of the continuum]].<ref>{{Harv|Robert|2000|loc=Chapter 1 Section 1.1}}</ref> For <math>\Z_p,</math> this results from the {{mvar|p}}-adic representation, which defines a [[bijection]] of <math>\Z_p</math> on the [[power set]] <math>\{0,\ldots,p-1\}^\N.</math> For <math>\Q_p</math> this results from its expression as a [[countably infinite]] [[union (set theory)|union]] of copies of <math>\Z_p</math>:
: <math>\Q_p=\bigcup_{i=0}^\infty \frac 1{p^i}\Z_p.</math>

== Algebraic closure ==

<math>\Q_p</math> contains <math>\Q</math> and is a field of [[characteristic (algebra)|characteristic]] {{math|0}}.

{{anchor|not_orderable}}Because {{math|0}} can be written as sum of squares,<ref>According to [[Hensel's lemma#Examples|Hensel's lemma]] <math>\Q_2</math> contains a square root of {{math|−7}}, so that <math>2^2 +1^2+1^2+1^2+\left(\sqrt{-7}\right)^2 = 0 ,</math> and if {{math|''p'' > 2}} then also by Hensel's lemma <math>\Q_p</math> contains a square root of {{math|1 − ''p''}}, thus 
<math>(p-1)\times 1^2 +\left(\sqrt{1-p}\right)^2 = 0 .</math></ref> <math>\Q_p</math>  cannot be turned into an [[Ordered field#Orderability of fields|ordered field]].

The field of [[real numbers]] <math>\R</math> has only a single proper [[algebraic extension]]: the [[complex numbers]] <math>\C</math>.  In other words, this [[quadratic extension]] is already [[algebraically closed field|algebraically closed]]. By contrast, the [[algebraic closure]] of <math>\Q_p</math>, denoted <math>\overline{\Q_p},</math> has infinite degree,<ref>{{Harv|Gouvêa|1997|loc=Corollary 5.3.10}}</ref> that is, <math>\Q_p</math> has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the {{mvar|p}}-adic valuation to <math>\overline{\Q_p},</math> the latter is not (metrically) complete.<ref>{{Harv|Gouvêa|1997|loc=Theorem 5.7.4}}</ref><ref name=C149>{{Harv|Cassels|1986|p=149}}</ref> Its (metric) completion is called <math>\C_p</math> or <math>\Omega_p</math>.<ref name=C149/><ref name=K13>{{Harv|Koblitz|1980|p=13}}</ref> Here an end is reached, as <math>\C_p</math> is algebraically closed.<ref name=C149/><ref>{{Harv|Gouvêa|1997|loc=Proposition 5.7.8}}</ref> However unlike <math>\C</math> this field is not [[locally compact]].<ref name=K13/>

<math>\C_p</math> and <math>\C</math> are isomorphic as rings,<ref>Two algebraically closed fields are isomorphic if and only if they have the same characteristic and transcendence degree (see, for example Lang’s ''Algebra'' X §1), and both <math>\C_p</math> and <math>\C</math> have characteristic zero and the cardinality of the continuum.</ref> so we may regard <math>\C_p</math> as <math>\C</math> 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 proof|constructive]]).

If <math>K</math> is any finite [[Galois extension]] of <math>\Q_p,</math>, the [[Galois group]] <math>\operatorname{Gal} \left(K/\Q_p \right)</math> is [[solvable group|solvable]]. Thus, the Galois group <math>\operatorname{Gal} \left(\overline{\Q_p}/ \Q_p \right)</math> is [[prosolvable]].

== Multiplicative group ==
<math>\Q_p</math> contains the {{mvar|n}}-th [[cyclotomic field]] ({{math|''n'' > 2}}) if and only if {{math|''n''&thinsp;{{!}} ''p'' − 1}}.<ref>{{Harv|Gouvêa|1997|loc=Proposition 3.4.2}}</ref> For instance, the {{mvar|n}}-th cyclotomic field is a subfield of <math>\Q_{13}</math> if and only if {{math|''n'' {{=}} 1, 2, 3, 4, 6}}, or {{math|12}}. In particular, there is no multiplicative {{mvar|p}}-[[torsion (algebra)|torsion]] in <math>\Q_p</math> if {{math|''p'' > 2}}. Also, {{math|−1}} is the only non-trivial torsion element in <math>\Q_2</math>.

Given a [[natural number]] {{mvar|k}}, the [[index (group theory)|index]] of the multiplicative group of the {{mvar|k}}-th powers of the non-zero elements of <math>\Q_p</math> in <math>\Q_p^\times</math> is finite.

The number {{mvar|[[e (mathematical constant)|e]]}}, defined as the sum of [[reciprocal (mathematics)|reciprocals]] of [[factorial]]s, is not a member of any {{mvar|p}}-adic field; but  <math>e^p \in \Q_p</math> for <math>p \ne 2</math>. For {{math|''p'' {{=}} 2}} one must take at least the fourth power.<ref>{{Harv|Robert|2000|loc=Section 4.1}}</ref> (Thus a number with similar properties as {{mvar|e}} — namely a {{mvar|p}}-th root of {{math|''e<sup>p</sup>''}} — is a member of  <math>\Q_p</math> for all {{mvar|p}}.)

== Local–global principle ==
[[Helmut Hasse]]'s [[Hasse principle|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 {{mvar|p}}-adic numbers for every prime&nbsp;{{mvar|p}}. This principle holds, for example, for equations given by [[quadratic form]]s, but fails for higher polynomials in several indeterminates.

== Rational arithmetic with Hensel lifting ==
{{main|Hensel lifting}}

== Generalizations and related concepts ==
The reals and the {{mvar|p}}-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general [[algebraic number field]]s, 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<sub>''P''</sub>(''x'') for the exponent of ''P'' in this factorization, and for any choice of number ''c'' greater than 1 we can set
:<math>|x|_P = c^{-\!\operatorname{ord}_P(x)}.</math>
Completing with respect to this absolute value {{nowrap begin}}|⋅|<sub>''P''</sub>{{nowrap end}} yields a field ''E''<sub>''P''</sub>, 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 [[absolute value (algebra)|non-Archimedean absolute value]] on ''E'' arises as some {{nowrap begin}}|⋅|<sub>''P''</sub>{{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'''<sub>''p''</sub>, 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 ring]]s and [[idele group]]s.

''p''-adic integers can be extended to [[Solenoid (mathematics)#p-adic solenoids|''p''-adic solenoids]] <math>\mathbb{T}_p</math>. There is a map from <math>\mathbb{T}_p</math> to the [[circle group]] whose fibers are the ''p''-adic integers <math>\mathbb{Z}_p</math>, in analogy to how there is a map from <math>\mathbb{R}</math> to the circle whose fibers are <math>\mathbb{Z}</math>.

== See also ==
{{div col}}
* [[Non-Archimedean (disambiguation)|Non-Archimedean]]
* [[p-adic quantum mechanics]]
* [[p-adic Hodge theory]]
* [[p-adic Teichmuller theory]]
* [[p-adic analysis]]
* [[p-adic valuation]]
* [[1 + 2 + 4 + 8 + ⋯]]
* [[Bijective numeration|''k''-adic notation]]
* [[C-minimal theory]]
* [[Hensel's lemma]]
* [[Locally compact field]]
* [[Mahler's theorem]]
* [[Profinite integer]]
* [[Volkenborn integral]]
* [[Two's complement]]
{{div col end}}

== Footnotes ==

=== Notes ===
{{reflist|group=note}}

=== Citations ===
{{reflist}}

== References ==
{{refbegin}}
* {{Citation |last=Cassels |first=J. W. S. |author-link=J. W. S. Cassels |title=Local Fields |series=London Mathematical Society Student Texts |volume=3 |publisher=[[Cambridge University Press]] |year=1986 |isbn=0-521-31525-5 |zbl=0595.12006}}
* {{citation|title=Theory of Algebraic Functions of One Variable|volume=39|series=History of mathematics|first1=Richard|last1=Dedekind|author1-link=Richard Dedekind|first2=Heinrich|last2=Weber|author2-link=Heinrich Martin Weber|publisher=American Mathematical Society|year=2012|isbn=978-0-8218-8330-3}}. &mdash; Translation into English by [[John Stillwell]] of ''Theorie der algebraischen Functionen einer Veränderlichen'' (1882).
* {{Citation|last=Gouvêa|first=F. Q.|date=March 1994|title=A Marvelous Proof|journal=[[American Mathematical Monthly]]|pages=203–222|volume=101|issue=3 |jstor=2975598|doi=10.2307/2975598}}
* {{Citation |last=Gouvêa |first=Fernando Q. |year=1997 |title=''p''-adic Numbers: An Introduction |edition=2nd |publisher=Springer |isbn=3-540-62911-4 | zbl=0874.11002}}
* {{citation|title=Handbook of Algebra|volume=6|editor-first=M.|editor-last=Hazewinkel|publisher=North Holland|date=2009|isbn=978-0-444-53257-2|page=342|url={{Google books|yimXZ-7L9ZoC|page=342|plainurl=yes}}}}
* {{Citation|last1=Hehner|first1=Eric C. R.|author-link1=Eric C. R. Hehner|last2=Horspool|first2=R. Nigel|year=1979|title=A new representation of the rational numbers for fast easy arithmetic|journal=[[SIAM Journal on Computing]]|pages=124–134 |volume=8 |issue=2 |doi=10.1137/0208011 |url=https://www.researchgate.net/publication/220617770|citeseerx=10.1.1.64.7714}}
* {{Citation | last = Hensel | first = Kurt | author-link=Kurt Hensel | title = Über eine neue Begründung der Theorie der algebraischen Zahlen | journal = Jahresbericht der Deutschen Mathematiker-Vereinigung | volume = 6 | year = 1897 | issue = 3 | pages = 83–88 | url = http://www.digizeitschriften.de/resolveppn/GDZPPN00211612X&L=2}}
* {{Citation|last1=Kelley|first1=John L.|author-link=John Leroy Kelley|title=General Topology|date=2008|orig-year=1955|publisher=Ishi Press|location=New York|isbn=978-0-923891-55-8}}
* {{Citation |last=Koblitz |first=Neal |author-link=Neal Koblitz |title=''p''-adic analysis: a short course on recent work |series=London Mathematical Society Lecture Note Series |volume=46 |publisher=[[Cambridge University Press]] |year=1980 |isbn=0-521-28060-5 |zbl=0439.12011}}
* {{Citation |last=Robert |first=Alain M. |year=2000 |title=A Course in ''p''-adic Analysis |publisher=Springer |isbn=0-387-98669-3}}
{{refend}}

== Further reading ==
{{refbegin}}
* {{Citation |last=Bachman |first=George |title=Introduction to ''p''-adic Numbers and Valuation Theory |year=1964 |publisher=Academic Press |isbn=0-12-070268-1}}
* {{Citation|last1=Borevich|first1=Z. I.|author-link1=Zenon Ivanovich Borevich|last2=Shafarevich|first2=I. R.|author2-link=Igor Rostislavovich Shafarevich|year=1986|title=Number Theory|publisher=Academic Press|location=Boston, MA|series=Pure and Applied Mathematics|volume=20|isbn=978-0-12-117851-2|url={{Google books|njgVUjjO-EAC|Number Theory|plainurl=yes}}|mr=0195803}}
* {{Citation |last=Koblitz |first=Neal |author-link=Neal Koblitz |year=1984 | series=[[Graduate Texts in Mathematics]] | volume=58 | title=''p''-adic Numbers, ''p''-adic Analysis, and Zeta-Functions | edition=2nd |publisher=Springer |isbn=0-387-96017-1}}
* {{Citation | last=Mahler | first=Kurt | author-link=Kurt Mahler | title=''p''-adic numbers and their functions | edition=2nd | zbl=0444.12013 | series=Cambridge Tracts in Mathematics | volume=76 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1981 | isbn=0-521-23102-7 | url-access=registration | url=https://archive.org/details/padicnumbersthei0000mahl }}
* {{Citation |last=Steen |first=Lynn Arthur |author-link=Lynn Arthur Steen |year=1978 |title=Counterexamples in Topology |publisher=Dover |isbn=0-486-68735-X|title-link=Counterexamples in Topology }}
{{refend}}

== External links ==
{{Commons category|P-adic numbers}}
* {{MathWorld|urlname=p-adicNumber|title=p-adic Number}}
* [http://www.encyclopediaofmath.org/index.php/P-adic_number ''p''-adic number] at [[Encyclopaedia of Mathematics|Springer On-line Encyclopaedia of Mathematics]]

{{Number systems}}
{{Authority control}}

{{DEFAULTSORT:P-Adic Number}}
[[Category:Field (mathematics)]]
[[Category:Number theory]]
[[Category:P-adic numbers| ]]