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 which 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 <math>\tfrac15</math> in [[Ternary numeral system|base Template:Math]] vs. the Template:Math-adic expansion, <math display="block">\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 Template:Mvar, a Template:Mvar-adic number can be defined as a series <math display="block">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 Template:Mvar is an integer (possibly negative), and each <math>a_i</math> is an integer such that <math>0\le a_i < p.</math> A Template:Mvar-adic integer is a Template:Mvar-adic number such that <math>k\ge 0.</math>

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]] <math>|s|_p=p^{-k},</math> where Template:Mvar is the least integer Template:Mvar such that <math>a_i\ne 0</math> (if all <math>a_i</math> 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,<ref>Template:Harv</ref> though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using Template:Mvar-adic numbers.<ref group="note">Translator's introduction, page 35: "Indeed, with hindsight it becomes apparent that a discrete valuation is behind Kummer's concept of ideal numbers." Template:Harv</ref>

MotivationEdit

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. If one knows that the absolute value of the result is less than Template:Mvar, this allows a computation of the result which does not involve any integer larger than Template:Mvar.

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 Template:Mvar, 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 Template:Mvar-adic number.

Basic lemmasEdit

The theory of Template:Mvar-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 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 the rational number and is called its Template:Mvar-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 Template:Mvar of valuation Template:Mvar can be uniquely written <math>r=ap^v+ s,</math> where Template:Mvar is a rational number of valuation greater than Template:Mvar, and Template:Mvar 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 Template:Mvar and Template:Mvar are integers coprime with Template:Mvar. By Bézout's lemma, there exist integers Template:Mvar and Template:Mvar, 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 display="block"> {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' = c/d</math>, where Template:Mvar is coprime to Template:Mvar; then <math>(a - a')d + p c = 0</math>, which is possible only if <math>a - a' = 0</math> and <math>c=0</math>. Hence <math>a = a'</math> and <math> s = s'</math>.

The above process can be iterated starting from Template:Mvar instead of Template:Mvar, giving the following.

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

The Template:Mvar-adic numbers are essentially obtained by continuing this infinitely to produce an infinite series.

p-adic seriesEdit

The Template:Mvar-adic numbers are commonly defined by means of Template:Mvar-adic series.

A Template:Mvar-adic series is a formal power series of the form <math display="block">\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 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 <math>p^k\tfrac nd,</math> with Template:Mvar and Template:Mvar both coprime with Template:Mvar.

Two Template:Mvar-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 Template:Mvar such that, for every integer <math>n>N,</math> the rational number <math display="block">\sum_{i=v}^n r_i p^{i} - \sum_{i=w}^n s_i p^{i} </math> is zero or has a Template:Mvar-adic valuation greater than Template:Mvar.

A Template:Mvar-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 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 <math>S\sim T,</math> one has <math display="block">\begin{align} S\pm U&\sim T\pm U,\\ SU&\sim TU,\\ 1/S&\sim 1/T. \end{align}</math>

The Template:Mvar-adic numbers are often defined as the equivalence classes of Template:Mvar-adic series, in a similar way as the definition of the real numbers as equivalence classes of Cauchy sequences. 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 <math>v_p(x)</math> is the exponent of Template:Mvar in the first non zero term of the corresponding normalized series; the valuation of zero is <math>v_p(0)=+\infty</math>
The Template:Mvar-adic absolute value of a nonzero Template:Mvar-adic number Template:Mvar, is <math>|x|_p=p^{-v(x)};</math> for the zero Template:Mvar-adic number, one has <math>|0|_p=0.</math>

Normalization of a p-adic seriesEdit

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 Template:Mvar-adic valuation of <math>r_v</math> is zero. For that, one considers the first nonzero <math>r_i.</math> 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 <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 Template:Math 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 Template:Math-adic series.

DefinitionEdit

There are several equivalent definitions of Template:Mvar-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use completion of a discrete valuation ring (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.

Template:Anchor With these operations, Template:Mvar-adic numbers form a field called the field of Template:Math-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 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 <math>v_p(x),</math> is the exponent of Template:Mvar in the first nonzero term of every Template:Mvar-adic series that represents Template:Mvar. 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 Template:Mvar-adic valuation of <math>\Q,</math> that is, the exponent Template:Mvar in the factorization of a rational number as <math dosplay=inline≝>\tfrac nd p^v,</math> with both Template:Mvar and Template:Mvar coprime with Template:Mvar.

p-adic integersEdit

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

A <math>p</math>-adic integer can be represented as a sequence <math display="block"> 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 Template:Mvar-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 Template:Mvar-adic series is the product of their first terms.
The units (invertible elements) of <math>\Z_p</math> 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 <math>\Z_{(p)} = \{\tfrac nd \mid n, d \in \Z,\, d \not\in p\Z \},</math> which is the localization of <math>\Z</math> at the prime ideal <math>p\Z.</math>

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 propertiesEdit

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 <math display="block">|x|_p = p^{-v_p(x)},</math> where <math>v_p(x)</math> is the Template:Mvar-adic valuation of Template:Mvar. The Template:Mvar-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 Template:Mvar and Template:Mvar 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 Template:Mvar-adic numbers a metric space, and even an ultrametric space, with the Template:Mvar-adic distance defined by <math>d_p(x,y)=|x-y|_p.</math>

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. 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 sums of a Template:Mvar-adic series, and thus a unique normalized Template:Mvar-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized Template:Mvar-adic series instead of equivalence classes of Cauchy sequences).

As the metric is defined from a discrete valuation, every open ball is also 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 Template:Mvar is the least integer such that <math>p^{-v}< r.</math> Similarly, <math>B_r[x] = B_{p^{-w}}(x),</math> where Template:Mvar is the greatest integer such that <math>p^{-w}>r.</math>

This implies that the Template:Mvar-adic numbers form a locally compact space (locally compact field), and the Template:Mvar-adic integers—that is, the ball <math>B_1[0]=B_p(0)</math>—form a compact space.

p-adic expansion of rational numbersEdit

The decimal expansion of a positive rational number <math>r</math> is its representation as a series <math display="block">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 Template:Mvar-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 integers both coprime with <math>p</math>, and <math>d</math> is positive. The integer <math>k</math> is the Template:Mvar-adic valuation of <math>r</math>, denoted <math>v_p(r),</math> and <math>p^{-k}</math> is its Template:Mvar-adic absolute value, denoted <math>|r|_p</math> (the absolute value is small when the valuation is large). The division step consists of writing Template:Anchor<math display="block">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 display="block">r = \sum_{i=k}^\infty a_i p^i</math> obtained by repeating indefinitely the above division step on successive remainders. In a Template:Mvar-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-Template:Mvar]].

The existence and the computation of the Template:Mvar-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 display="block">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 display="block">n t=q p+a,</math> with <math>0\le a <p.</math> This gives the division step as <math display="block">\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 display="block">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 Template:Mvar-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 Template:Mvar-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 Template:Mvar-adic absolute value. In the standard Template:Mvar-adic notation, the digits are written in the same order as in a [[Positional notation#Base of the numeral system|standard base-Template:Mvar 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 Template:Mvar-adic expansion of a rational number is eventually periodic. 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 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.

ExampleEdit

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 display="block">\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 "shift" of the Template:Mvar-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 display="block">-\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 display="block">-\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 display="block">\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 display="block">-\frac 23=1-\frac 53,</math> and <math display="block">\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 odd powers of five, and <math>1</math> for even powers. Or in the standard 5-adic notation <math display="block">\frac 13= \ldots 1313132_5 </math> with the ellipsis <math> \ldots </math> on the left hand side.

Positional notationEdit

It is possible to use a positional notation similar to that which is used to represent numbers in base Template:Mvar.

Let <math display = inline>\sum_{i=k}^\infty a_i p^i</math> be a normalized Template:Mvar-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 Template:Mvar, often with Template:Mvar appearing on the right as an index: <math display="block">\ldots a_n \ldots a_1{a_0}_p</math> So, the computation of the example above shows that <math display="block">\frac 13= \ldots 1313132_5,</math> and <math display="block">\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 Template:Mvar is present, it appears just after the separating dot. For example, <math display="block">\frac 1{15}= \ldots 3131313._52,</math> and <math display="block">\frac 1{75}= \ldots 1313131._532.</math>

If a Template:Mvar-adic representation is finite on the left (that is, <math>a_i=0</math> for large values of Template:Mvar), 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 base Template:Mvar. For these rational numbers, the two representations are the same.

Modular propertiesEdit

The quotient ring <math>\Z_p/p^n\Z_p</math> may be identified with the ring <math>\Z/p^n\Z</math> of the integers modulo <math>p^n.</math> This can be shown by remarking that every Template:Mvar-adic integer, represented by its normalized Template:Mvar-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 Template:Mvar.

The mapping that maps a normalized Template:Mvar-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 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 <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 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 <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 Template:Mvar of a polynomial with integer coefficients to a factorization modulo <math display = inline>p^n</math> for large values of Template:Mvar. This is commonly used by polynomial factorization algorithms.

NotationEdit

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 <math>\tfrac15,</math> for example, is written as <math display="block">\frac15 = \dots 121012102_3.</math>

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 <math>\tfrac15</math> is <math display="block"> \frac15 = 2.01210121\dots_3 \mbox{ or } \frac1{15} = 20.1210121\dots_3. </math>

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 <math>\tfrac15</math> can be written using balanced ternary digits Template:Math}, with Template:Math representing negative one, as <math display="block">\frac15 = \dots\underline{1}11\underline{11}11\underline{11}11\underline{1}_{\text{3}} .</math>

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.<ref>Template:Harv</ref>

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.<ref>Template:Harv</ref>

CardinalityEdit

Both <math>\Z_p</math> and <math>\Q_p</math> are uncountable and have the cardinality of the continuum.<ref>Template:Harv</ref> For <math>\Z_p,</math> this results from the Template:Mvar-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 of copies of <math>\Z_p</math>: <math display="block">\Q_p=\bigcup_{i=0}^\infty \frac 1{p^i}\Z_p.</math>

Algebraic closureEdit

<math>\Q_p</math> contains <math>\Q</math> and is a field of characteristic Template:Math.

Template:AnchorBecause Template:Math can be written as sum of squares,<ref>According to Hensel's lemma <math>\Q_2</math> contains a square root of Template:Math, so that <math>2^2 +1^2+1^2+1^2+\left(\sqrt{-7}\right)^2 = 0 ,</math> and if Template:Math then also by Hensel's lemma <math>\Q_p</math> contains a square root of Template:Math, 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.

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. By contrast, the algebraic closure of <math>\Q_p</math>, denoted <math>\overline{\Q_p},</math> has infinite degree,<ref>Template:Harv</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 Template:Mvar-adic valuation to <math>\overline{\Q_p},</math> the latter is not (metrically) complete.<ref>Template:Harv</ref><ref name=C149>Template:Harv</ref> Its (metric) completion is called <math>\C_p</math> or <math>\Omega_p</math>.<ref name=C149/><ref name=K13>Template:Harv</ref> Here an end is reached, as <math>\C_p</math> is algebraically closed.<ref name=C149/><ref>Template:Harv</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).

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. Thus, the Galois group <math>\operatorname{Gal} \left(\overline{\Q_p}/ \Q_p \right)</math> is prosolvable.

Multiplicative groupEdit

<math>\Q_p</math> contains the Template:Mvar-th cyclotomic field (Template:Math) if and only if Template:Math.<ref>Template:Harv</ref> For instance, the Template:Mvar-th cyclotomic field is a subfield of <math>\Q_{13}</math> if and only if Template:Math, or Template:Math. In particular, there is no multiplicative Template:Mvar-torsion in <math>\Q_p</math> if Template:Math. Also, Template:Math is the only non-trivial torsion element in <math>\Q_2</math>.

Given a natural number Template:Mvar, the index of the multiplicative group of the Template:Mvar-th powers of the non-zero elements of <math>\Q_p</math> in <math>\Q_p^\times</math> 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 <math>e^p \in \Q_p</math> for <math>p \ne 2</math>. For Template:Math one must take at least the fourth power.<ref>Template:Harv</ref> (Thus a number with similar properties as Template:Mvar — namely a Template:Mvar-th root of Template:Math — is a member of <math>\Q_p</math> for all Template:Mvar.)

Local–global principleEdit

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.

Rational arithmetic with Hensel liftingEdit

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Generalizations and related conceptsEdit

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 <math display="block">|x|_P = c^{-\!\operatorname{ord}_P(x)}.</math> 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 <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>.