Number Line Of Rational Numbers

Number system for a prime p which extends the rationals, defining closeness differently

The three-adic integers, with selected corresponding characters on their Pontryagin dual grouping

In mathematics, the p-adic number organisation for whatever prime numberp extends the ordinary arithmetics of the rational numbers in a different style from the extension of the rational number organisation to the existent and complex number systems. The extension is accomplished past an alternative interpretation of the concept of "closeness" or absolute value. In item, 2 p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to accept powerful applications in number theory – including, for example, in the famous proof of Fermat's Concluding Theorem by Andrew Wiles.^[1]

These numbers were first described by Kurt Hensel in 1897,^[2] though, with hindsight, some of Ernst Kummer'southward earlier work can exist interpreted as implicitly using p-adic numbers.^{[annotation i]} The p-adic numbers were motivated primarily by an try to bring the ideas and techniques of power serial methods into number theory. Their influence now extends far across this. For example, the field of p-adic analysis essentially provides an alternative course of calculus.

More formally, for a given primep, the field Q _p of p-adic numbers is a completion of the rational numbers. The field Q _p is also given a topology derived from a metric, which is itself derived from the p -adic club, an culling valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a signal in Q _p . This is what allows the evolution of calculus on Q _p , and it is the interaction of this analytic and algebraic construction that gives the p-adic number systems their power and utility.

The p in "p-adic" is a variable and may exist replaced with a prime (yielding, for case, "the 2-adic numbers") or another expression representing a prime number. The "adic" of "p-adic" comes from the catastrophe plant in words such as dyadic or triadic.

p-adic expansion of rational numbers [edit]

The decimal expansion of a positive rational number ${\displaystyle r}$ is its representation equally a series

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

where ${\displaystyle k}$ is an integer and each ${\displaystyle a_{i}}$ is too an integer such that ${\displaystyle 0\leq a_{i}<10.}$ This expansion tin can be computed past long segmentation of the numerator by the denominator, which is itself based on the post-obit theorem: If ${\displaystyle r={\tfrac {n}{d}}}$ is a rational number such that ${\displaystyle x^{k}\leq r<10^{k+1},}$ there is an integer ${\displaystyle a}$ such that ${\displaystyle 0<a<10,}$ and ${\displaystyle r=a\,10^{k}+r',}$ with ${\displaystyle r'<x^{one thousand}.}$ The decimal expansion is obtained by repeatedly applying this result to the remainder ${\displaystyle r'}$ which in the iteration assumes the role of the original rational number ${\displaystyle r}$ .

The p-adic expansion of a rational number is defined similarly, just with a different segmentation step. More precisely, given a fixed prime number ${\displaystyle p}$ , every nonzero rational number ${\displaystyle r}$ tin be uniquely written as ${\displaystyle r=p^{k}{\tfrac {northward}{d}},}$ where ${\displaystyle k}$ is a (possibly negative) integer, ${\displaystyle n}$ and ${\displaystyle d}$ are coprime integers both coprime with ${\displaystyle p}$ , and ${\displaystyle d}$ is positive. The integer ${\displaystyle k}$ is the p-adic valuation of ${\displaystyle r}$ , denoted ${\displaystyle v_{p}(r),}$ and ${\displaystyle p^{-k}}$ is its p-adic absolute value, denoted ${\displaystyle |r|_{p}}$ (the absolute value is modest when the valuation is large). The division footstep consists of writing

${\displaystyle r=a\,p^{1000}+r'}$

where ${\displaystyle a}$ is an integer such that ${\displaystyle 0\leq a<p,}$ and ${\displaystyle r'}$ is either zero, or a rational number such that ${\displaystyle |r'|_{p}<p^{-k}}$ (that is, ${\displaystyle v_{p}(r')>1000}$ ).

The ${\displaystyle p}$ -adic expansion of ${\displaystyle r}$ is the formal power series

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

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

If ${\displaystyle r=p^{k}{\tfrac {n}{ane}}}$ with ${\displaystyle n>0}$ , the process stops somewhen with a goose egg residue; in this instance, the serial is completed by abaft terms with a zero coefficient, and is the representation of ${\displaystyle r}$ in base-p.

The being and the computation of the p-adic expansion of a rational number results from Bézout's identity in the following way. If, as to a higher place, ${\displaystyle r=p^{k}{\tfrac {n}{d}},}$ and ${\displaystyle d}$ and ${\displaystyle p}$ are coprime, there exist integers ${\displaystyle t}$ and ${\displaystyle u}$ such that ${\displaystyle td+up=i.}$ And so

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

And then, the Euclidean segmentation of ${\displaystyle nt}$ by ${\displaystyle p}$ gives

${\displaystyle nt=qp+a,}$

with ${\displaystyle 0\leq a<p.}$ This gives the division stride as

${\displaystyle {\begin{array}{lcl}r&=&p^{thou}(qp+a)+p^{1000+ane}{\frac {united nations}{d}}\\&=&ap^{k}+p^{thou+ane}\,{\frac {qd+un}{d}},\\\end{array}}}$

so that in the iteration

${\displaystyle r'=p^{thou+one}\,{\frac {qd+united nations}{d}}}$

is the new rational number.

The uniqueness of the division step and of the whole p-adic expansion is like shooting fish in a barrel: if ${\displaystyle p^{1000}a_{1}+p^{k+ane}s_{1}=p^{chiliad}a_{2}+p^{chiliad+i}s_{2},}$ 1 has ${\displaystyle a_{one}-a_{ii}=p(s_{2}-s_{1}).}$ This means ${\displaystyle p}$ divides ${\displaystyle a_{one}-a_{2}.}$ Since ${\displaystyle 0\leq a_{1}<p}$ and ${\displaystyle 0\leq a_{ii}<p,}$ the following must be true: ${\displaystyle 0\leq a_{1}}$ and ${\displaystyle a_{ii}<p.}$ Thus, i gets ${\displaystyle -p<a_{one}-a_{2}<p,}$ and since ${\displaystyle p}$ divides ${\displaystyle a_{1}-a_{2}}$ information technology must be that ${\displaystyle a_{1}=a_{ii}.}$

The 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 p-adic absolute value. In the standard p-adic annotation, the digits are written in the same club as in a standard base-p system, namely with the powers of the base increasing to the left. This means that the product of the digits is reversed and the limit happens on the left hand side.

The p-adic expansion of a rational number is somewhen periodic. Conversely, a serial ${\textstyle \sum _{i=k}^{\infty }a_{i}p^{i},}$ with ${\displaystyle 0\leq a_{i}<p}$ converges (for the p-adic accented value) to a rational number if and just if it is somewhen periodic; in this case, the series is the p-adic expansion of that rational number. The proof is similar to that of the similar consequence for repeating decimals.

Case [edit]

Let us compute the 5-adic expansion of ${\displaystyle {\frac {i}{3}}.}$ Bézout's identity for v and the denominator 3 is ${\displaystyle 2\cdot 3+(-1)\cdot five=i}$ (for larger examples, this tin can be computed with the extended Euclidean algorithm). Thus

${\displaystyle {\frac {1}{3}}=2-{\frac {5}{3}}.}$

For the side by side step, one has to "divide" ${\displaystyle -1/3}$ (the gene 5 in the numerator of the fraction has to be viewed every bit a "shift" of the p-adic valuation, and thus it is not involved in the "division"). Multiplying Bézout's identity past ${\displaystyle -i}$ gives

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

The "integer part" ${\displaystyle -2}$ is not in the right interval. Then, ane has to utilise Euclidean division by ${\displaystyle v}$ for getting ${\displaystyle -ii=three-1\cdot 5,}$ giving

${\displaystyle -{\frac {1}{three}}=3-5+{\frac {5}{iii}}=three-{\frac {10}{3}},}$

and

${\displaystyle {\frac {one}{iii}}=two+iii\cdot v+{\frac {-2}{3}}\cdot 5^{2}.}$

Similarly, one has

${\displaystyle -{\frac {ii}{3}}=i-{\frac {5}{3}},}$

and

${\displaystyle {\frac {1}{3}}=two+3\cdot 5+1\cdot 5^{2}+{\frac {-1}{3}}\cdot v^{three}.}$

As the "rest" ${\displaystyle -{\tfrac {one}{3}}}$ has already been institute, the process can be connected easily, giving coefficients ${\displaystyle 3}$ for odd powers of five, and ${\displaystyle 1}$ for even powers. Or in the standard 5-adic notation

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

with the ellipsis ${\displaystyle \ldots }$ on the left hand side.

p-adic series [edit]

In this commodity, given a prime number p, a p-adic series is a formal serial of the form

${\displaystyle \sum _{i=grand}^{\infty }a_{i}p^{i},}$

where every nonzero ${\displaystyle a_{i}}$ is a rational number ${\displaystyle a_{i}={\tfrac {n_{i}}{d_{i}}},}$ such that none of ${\displaystyle n_{i}}$ and ${\displaystyle d_{i}}$ is divisible by p.

Every rational number may exist viewed every bit a p-adic serial with a single term, consisting of its factorization of the grade ${\displaystyle p^{k}{\tfrac {n}{d}},}$ with n and d both coprime with p.

A p-adic serial is normalized if each ${\displaystyle a_{i}}$ is an integer in the interval ${\displaystyle [0,p-one].}$ So, the p-adic expansion of a rational number is a normalized p-adic series.

The p-adic valuation, or p-adic order of a nonzero p-adic serial is the everyman integer i such that ${\displaystyle a_{i}\neq 0.}$ The order of the goose egg series is infinity ${\displaystyle \infty .}$

Ii p-adic serial are equivalent if they have the aforementioned order k, and if for every integer n ≥ k the difference between their partial sums

${\displaystyle \sum _{i=yard}^{n}a_{i}p^{i}-\sum _{i=k}^{n}b_{i}p^{i}=\sum _{i=g}^{n}(a_{i}-b_{i})p^{i}}$

has an guild greater than n (that is, is a rational number of the class ${\displaystyle p^{k}{\tfrac {a}{b}},}$ with ${\displaystyle k>northward,}$ and a and b both coprime with p).

For every p-adic series ${\displaystyle S}$ , there is a unique normalized series ${\displaystyle N}$ such that ${\displaystyle S}$ and ${\displaystyle Due north}$ are equivalent. ${\displaystyle Northward}$ is the normalization of ${\displaystyle S.}$ The proof is similar to the beingness proof of the p-adic expansion of a rational number. In particular, every rational number can be considered every bit a p-adic serial with a single nonzero term, and the normalization of this series is exactly the rational representation of the rational number.

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

The usual operations of series (add-on, subtraction, multiplication, segmentation) map p-adic series to p-adic serial, and are compatible with equivalence of p-adic series. That is, denoting the equivalence with ~, if S, T and U are nonzero p-adic serial such that ${\displaystyle Southward\sim T,}$ one has

${\displaystyle {\begin{aligned}S\pm U&\sim T\pm U,\\SU&\sim TU,\\1/Due south&\sim i/T.\end{aligned}}}$

Moreover, Southward and T have the aforementioned order, and the aforementioned first term.

Positional notation [edit]

It is possible to use a positional notation similar to that which is used to stand for numbers in base p.

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

If ${\displaystyle thou\geq 0,}$ the positional annotation consists of writing the ${\displaystyle a_{i}}$ consecutively, ordered by decreasing values of i, ofttimes with p appearing on the right as an alphabetize:

${\displaystyle \ldots a_{n}\ldots a_{one}{a_{0}}_{p}}$

So, the computation of the example above shows that

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

and

${\displaystyle {\frac {25}{three}}=\ldots 131313200_{5}.}$

When ${\displaystyle k<0,}$ a separating dot is added before the digits with negative index, and, if the index p is present, information technology appears merely after the separating dot. For instance,

${\displaystyle {\frac {1}{xv}}=\ldots 3131313._{5}2,}$

and

${\displaystyle {\frac {one}{75}}=\ldots 1313131._{v}32.}$

If a p-adic representation is finite on the left (that is, ${\displaystyle a_{i}=0}$ for large values of i), and then it has the value of a nonnegative rational number of the class ${\displaystyle np^{five},}$ with ${\displaystyle n,v}$ integers. These rational numbers are exactly the nonnegative rational numbers that take a finite representation in base p. For these rational numbers, the ii representations are the same.

Definition [edit]

There are several equivalent definitions of p-adic numbers. The one that is given here is relatively elementary, since it does not involve whatsoever other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use completion of a discrete valuation ring (run into §p-adic integers), completion of a metric space (encounter § Topological properties), or inverse limits (see § Modular properties).

A p-adic number can exist divers as a normalized p-adic series. Since there are other equivalent definitions that are commonly used, one says oft that a normalized p-adic serial represents a p-adic number, instead of proverb that it is a p-adic number.

One tin can say as well that whatsoever p-adic serial represents a p-adic number, since every p-adic series is equivalent to a unique normalized p-adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of p-adic numbers: the issue of such an operation is obtained past normalizing the result of the respective functioning on series. This well defines operations on p-adic numbers, since the series operations are compatible with equivalence of p-adic series.

With these operations, p-adic numbers grade a field called the field of p -adic numbers and denoted ${\displaystyle \mathbb {Q} _{p}}$ or ${\displaystyle \mathbf {Q} _{p}.}$ In that location is a unique field homomorphism from the rational numbers into the p-adic numbers, which maps a rational number to its p-adic expansion. The image of this homomorphism is commonly identified with the field of rational numbers. This allows considering the p -adic numbers every bit an extension field of the rational numbers, and the rational numbers as a subfield of the p -adic numbers.

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

p-adic integers [edit]

The p-adic integers are the p-adic numbers with a nonnegative valuation.

A p-adic integer tin can be represented equally a sequence

${\displaystyle 10=(x_{1}\operatorname {modern} p,~x_{2}\operatorname {mod} p^{2},~x_{iii}\operatorname {mod} p^{3},~\ldots )}$

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

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

The p-adic integers class a commutative ring, denoted ${\displaystyle \mathbb {Z} _{p}}$ or ${\displaystyle \mathbf {Z} _{p}}$ , that has the following properties.

• It is an integral domain, since it is a subring of a field, or since the first term of the serial representation of the product of two not cypher p-adic series is the product of their first terms.
• The units (invertible elements) of ${\displaystyle \mathbb {Z} _{p}}$ are the p-adic numbers of valuation zero.
• Information technology is a principal ideal domain, such that each ideal is generated past a ability of p.
• Information technology is a local ring of Krull dimension one, since its but prime number ideals are the zero platonic and the ideal generated past p, the unique maximal ideal.
• Information technology is a discrete valuation band, since this results from the preceding properties.
• Information technology is the completion of the local band ${\displaystyle \mathbb {Z} _{(p)}=\{{\tfrac {northward}{d}}\mid n,d\in \mathbb {Z} ,\,d\not \in p\mathbb {Z} \},}$ which is the localization of ${\displaystyle \mathbb {Z} }$ at the prime ideal ${\displaystyle p\mathbb {Z} .}$

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

Topological properties [edit]

The p-adic valuation allows defining an absolute value on p-adic numbers: the p-adic absolute value of a nonzero p-adic number x is

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

where ${\displaystyle v_{p}(x)}$ is the p-adic valuation of x. The p-adic accented value of ${\displaystyle 0}$ is ${\displaystyle |0|_{p}=0.}$ This is an absolute value that satisfies the stiff triangle inequality since, for every x and y 1 has

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

This makes the p-adic numbers a metric space, and even an ultrametric space, with the p-adic distance defined by ${\displaystyle d_{p}(x,y)=|x-y|_{p}.}$

Every bit a metric space, the p-adic numbers form the completion of the rational numbers equipped with the p-adic accented value. This provides another way for defining the p-adic numbers. Notwithstanding, the general construction of a completion tin can exist simplified in this case, because the metric is defined past a discrete valuation (in short, ane 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 fractional sums of a p-adic serial, and thus a unique normalized p-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized p-adic series instead of equivalence classes of Cauchy sequences).

As the metric is defined from a detached valuation, every open ball is likewise closed. More precisely, the open brawl ${\displaystyle B_{r}(10)=\{y\mid d_{p}(x,y)<r\}}$ equals the closed ball ${\displaystyle B_{p^{-v}}[x]=\{y\mid d_{p}(x,y)\leq p^{-5}\},}$ where 5 is the least integer such that ${\displaystyle p^{-five}<r.}$ Similarly, ${\displaystyle B_{r}[x]=B_{p^{-w}}(x),}$ where w is the greatest integer such that ${\displaystyle p^{-w}>r.}$

This implies that the p-adic numbers grade a locally compact infinite, and the p-adic integers—that is, the brawl ${\displaystyle B_{1}[0]=B_{p}(0)}$ —class a compact space.

Modular properties [edit]

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

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

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

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

For case, for computing the p-adic (multiplicative) inverse of an integer, 1 can use Newton's method, starting from the inverse modulo p; then, each Newton pace computes the inverse modulo ${\textstyle p^{n^{2}}}$ from the changed modulo ${\textstyle p^{due north}.}$

The aforementioned method can exist used for computing the p-adic square root of an integer that is a quadratic residue modulo p. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to exam whether the given integer is the square of the value found in ${\displaystyle \mathbb {Z} _{p}/p^{northward}\mathbb {Z} _{p}}$ . Applying Newton's method to discover the square root requires ${\textstyle p^{n}}$ to be larger than twice the given integer, which is speedily satisfied.

Hensel lifting is a like method that allows to "lift" the factorization modulo p of a polynomial with integer coefficients to a factorization modulo ${\textstyle p^{due north}}$ for big values of n. This is commonly used by polynomial factorization algorithms.

Notation [edit]

There are several different conventions for writing p-adic expansions. So far this article has used a notation for p-adic expansions in which powers ofp increase from right to left. With this right-to-left notation the 3-adic expansion of 1⁄5 , for example, is written every bit

${\displaystyle {\dfrac {one}{5}}=\dots 121012102_{3}.}$

When performing arithmetics in this notation, digits are carried to the left. It is as well possible to write p-adic expansions so that the powers of p increase from left to correct, and digits are carried to the right. With this left-to-right note the 3-adic expansion of ane⁄five is

${\displaystyle {\dfrac {one}{5}}=2.01210121\dots _{3}{\mbox{ or }}{\dfrac {1}{15}}=20.1210121\dots _{3}.}$

p-adic expansions may be written with other sets of digits instead of {0, i, ..., p − one}. For case, the 3-adic expansion of ¹/₅ can be written using counterbalanced ternary digits {1,0,ane} as

${\displaystyle {\dfrac {1}{5}}=\dots {\underline {1}}11{\underline {11}}11{\underline {11}}eleven{\underline {one}}_{\text{iii}}.}$

In fact any set of p integers which are in distinct balance classes modulo p may be used as p-adic digits. In number theory, Teichmüller representatives are sometimes used as digits.^[iii]

Quote note is a variant of the p-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.^[four]

Cardinality [edit]

Both ${\displaystyle \mathbb {Z} _{p}}$ and ${\displaystyle \mathbb {Q} _{p}}$ are uncountable and have the cardinality of the continuum.^[5] For ${\displaystyle \mathbb {Z} _{p},}$ this results from the p-adic representation, which defines a bijection of ${\displaystyle \mathbb {Z} _{p}}$ on the power set ${\displaystyle \{0,\ldots ,p-i\}^{\mathbb {N} }.}$ For ${\displaystyle \mathbb {Q} _{p}}$ this results from its expression as a countably space union of copies of ${\displaystyle \mathbb {Z} _{p}}$ :

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

Algebraic closure [edit]

Q _p contains Q and is a field of feature 0.

Because 0 can be written as sum of squares,^[six] Q _p cannot be turned into an ordered field.

R has just a single proper algebraic extension: C ; in other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of Q _p , denoted ${\displaystyle {\overline {\mathbf {Q} _{p}}},}$ has infinite caste,^[vii] that is, Q _p has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the p-adic valuation to ${\displaystyle {\overline {\mathbf {Q} _{p}}},}$ the latter is not (metrically) complete.^{[viii] [9]} Its (metric) completion is called C _p or Ω _p .^{[9] [x]} Hither an end is reached, as C _p is algebraically closed.^{[nine] [eleven]} However different C this field is not locally compact.^[10]

C _p and C are isomorphic every bit rings, so we may regard C _p as C endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the axiom of choice, and does not provide an explicit example of such an isomorphism (that is, it is not constructive).

If K is a finite Galois extension of Q _p , the Galois group ${\displaystyle \operatorname {Gal} \left(\mathbf {K} /\mathbf {Q} _{p}\correct)}$ is solvable. Thus, the Galois group ${\displaystyle \operatorname {Gal} \left({\overline {\mathbf {Q} _{p}}}/\mathbf {Q} _{p}\right)}$ is prosolvable.

Multiplicative group [edit]

Q _p contains the due north-thursday cyclotomic field ( north > 2) if and only if due north | p − 1.^[12] For case, the n-th cyclotomic field is a subfield of Q ₁₃ if and only if n = 1, ii, iii, 4, 6, or 12. In particular, there is no multiplicative p-torsion in Q _p , if p > 2. Also, −1 is the just non-trivial torsion element in Q ₂ .

Given a natural number m, the alphabetize of the multiplicative group of the k-thursday powers of the non-zero elements of Q _p in ${\displaystyle \mathbf {Q} _{p}^{\times }}$ is finite.

The number e, defined as the sum of reciprocals of factorials, is not a fellow member of any p-adic field; but east ^p ∈ Q _p (p ≠ 2). For p = two one must have at least the fourth power.^[13] (Thus a number with similar backdrop as e — namely a p-th root of e^p — is a member of ${\displaystyle {\overline {\mathbf {Q} _{p}}}}$ for all p.)

Local–global principle [edit]

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

Rational arithmetics with Hensel lifting [edit]

Generalizations and related concepts [edit]

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

Suppose D is a Dedekind domain and E is its field of fractions. Pick a non-zero prime number platonic P of D. If x is a non-zero chemical element of Due east, then xD is a fractional platonic and can be uniquely factored as a production of positive and negative powers of not-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 i nosotros tin set

${\displaystyle |10|_{P}=c^{-\!\operatorname {ord} _{P}(10)}.}$

Completing with respect to this absolute value | . | _P yields a field E _P , the proper generalization of the field of p-adic numbers to this setting. The choice of c does not modify the completion (different choices yield the same concept of Cauchy sequence, so the aforementioned completion). Information technology is convenient, when the balance field D/P is finite, to take for c the size of D/P.

For example, when East is a number field, Ostrowski'due south theorem says that every non-trivial non-Archimedean accented value on Eastward arises equally some | . | _P . The remaining non-trivial absolute values on E arise from the unlike embeddings of E into the real or complex numbers. (In fact, the non-Archimedean absolute values tin can be considered as simply the dissimilar embeddings of E into the fields C _p , thus putting the description of all the non-niggling absolute values of a number field on a common footing.)

Frequently, one needs to simultaneously keep track of all the higher up-mentioned completions when E is a number field (or more more often than not a global field), which are seen as encoding "local" information. This is achieved by adele rings and idele groups.

p-adic integers tin can be extended to p-adic solenoids ${\displaystyle \mathbb {T} _{p}}$ . In that location is a map from ${\displaystyle \mathbb {T} _{p}}$ to the circumvolve group whose fibers are the p-adic integers ${\displaystyle \mathbb {Z} _{p}}$ , in analogy to how at that place is a map from ${\displaystyle \mathbb {R} }$ to the circle whose fibers are ${\displaystyle \mathbb {Z} }$ .

See too [edit]

• Not-archimedean
• p-adic quantum mechanics
• p-adic Hodge theory
• p-adic Teichmuller theory
• p-adic assay
• 1 + 2 + 4 + 8 + ...
• 1000-adic notation
• C-minimal theory
• Hensel's lemma
• Locally compact field
• Mahler's theorem
• Profinite integer
• Volkenborn integral

Footnotes [edit]

Notes [edit]

1. ^ Translator's introduction, page 35: "Indeed, with hindsight it becomes apparent that a discrete valuation is behind Kummer's concept of platonic numbers."(Dedekind & Weber 2012, p. 35)

Citations [edit]

1. ^ (Gouvêa 1994, pp. 203–222)
2. ^ (Hensel 1897)
3. ^ (Hazewinkel 2009, p. 342)
4. ^ (Hehner & Horspool 1979, pp. 124–134)
5. ^ (Robert 2000, Chapter 1 Section 1.1)
6. ^ According to Hensel's lemma Q ₂ contains a square root of −vii, so that ${\displaystyle 2^{2}+1^{2}+1^{2}+1^{2}+\left({\sqrt {-7}}\right)^{2}=0,}$ and if p > 2 so besides by Hensel's lemma Q _p contains a foursquare root of 1 − p , thus ${\displaystyle (p-ane)\times i^{two}+\left({\sqrt {one-p}}\correct)^{2}=0.}$
7. ^ (Gouvêa 1997, Corollary 5.3.10)
8. ^ (Gouvêa 1997, Theorem v.7.4)
9. ^ ^{a b c} (Cassels 1986, p. 149)
10. ^ ^{a b} (Koblitz 1980, p. 13)
11. ^ (Gouvêa 1997, Proposition 5.seven.8)
12. ^ (Gouvêa 1997, Proposition iii.iv.2)
13. ^ (Robert 2000, Section iv.ane)

References [edit]

• Cassels, J. Westward. S. (1986), Local Fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, ISBN0-521-31525-5, Zbl 0595.12006
• Dedekind, Richard; Weber, Heinrich (2012), Theory of Algebraic Functions of One Variable, History of mathematics, vol. 39, American Mathematical Society, ISBN978-0-8218-8330-3 . — Translation into English past John Stillwell of Theorie der algebraischen Functionen einer Veränderlichen (1882).
• Gouvêa, F. Q. (March 1994), "A Marvelous Proof", American Mathematical Monthly, 101 (iii): 203–222, doi:10.2307/2975598, JSTOR 2975598
• Gouvêa, Fernando Q. (1997), p-adic Numbers: An Introduction (2nd ed.), Springer, ISBNiii-540-62911-iv, Zbl 0874.11002
• Hazewinkel, M., ed. (2009), Handbook of Algebra, vol. half dozen, North Holland, p. 342, ISBN978-0-444-53257-2
• Hehner, Eric C. R.; Horspool, R. Nigel (1979), "A new representation of the rational numbers for fast easy arithmetic", SIAM Journal on Computing, viii (2): 124–134, CiteSeerXx.ane.one.64.7714, doi:10.1137/0208011
• Hensel, Kurt (1897), "Über eine neue Begründung der Theorie der algebraischen Zahlen", Jahresbericht der Deutschen Mathematiker-Vereinigung, six (three): 83–88
• Kelley, John L. (2008) [1955], General Topology, New York: Ishi Press, ISBN978-0-923891-55-viii
• Koblitz, Neal (1980), p-adic assay: a brusk course on recent work, London Mathematical Guild Lecture Note Serial, vol. 46, Cambridge University Printing, ISBN0-521-28060-5, Zbl 0439.12011
• Robert, Alain M. (2000), A Form in p-adic Analysis, Springer, ISBN0-387-98669-iii

Further reading [edit]

• Bachman, George (1964), Introduction to p-adic Numbers and Valuation Theory, Academic Press, ISBN0-12-070268-i
• Borevich, Z. I.; Shafarevich, I. R. (1986), Number Theory, Pure and Applied Mathematics, vol. 20, Boston, MA: Bookish Press, ISBN978-0-12-117851-2, MR 0195803
• Koblitz, Neal (1984), p-adic Numbers, p-adic Assay, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58 (2nd ed.), Springer, ISBN0-387-96017-1
• Mahler, Kurt (1981), p-adic numbers and their functions , Cambridge Tracts in Mathematics, vol. 76 (2nd ed.), Cambridge: Cambridge University Printing, ISBN0-521-23102-7, Zbl 0444.12013
• Steen, Lynn Arthur (1978), Counterexamples in Topology, Dover, ISBN0-486-68735-X

External links [edit]

• Weisstein, Eric W. "p-adic Number". MathWorld.
• p-adic number at Springer On-line Encyclopaedia of Mathematics
• Completion of Algebraic Closure – on-line lecture notes by Brian Conrad
• An Introduction to p-adic Numbers and p-adic Analysis - on-line lecture notes by Andrew Baker, 2007
• Efficient p-adic arithmetic (slides)
• Introduction to p-adic numbers
• Houston-Edwards, Kelsey (October 19, 2020), An Space Universe of Number Systems, Quanta Magazine

Number Line Of Rational Numbers,

Source: https://en.wikipedia.org/wiki/P-adic_number

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