''p''-adic expansion of rational numbers
and r' is either zero, or a rational number such that , r', _p < p^ (that is, v_p(r')>k). The p-''adic expansion'' of r is the
and 0\le a_2 the following must be true: 0\le a_1 and a_2 Thus, one gets -p < a_1-a_2 < p, and since p divides a_1-a_2 it must be that a_1=a_2. The -adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a convergent series In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted :S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k. The th partial sum ... with the -adic absolute value. In the standard -adic notation, the digits are written in the same order as in a standard base- 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 -adic expansion of a rational number is eventually periodic. Conversely, a series \sum_^\infty a_i p^i, with 0\le a_i converges (for the -adic absolute value) to a rational number if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ... it is eventually periodic; in this case, the series is the -adic expansion of that rational number. The proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ... is similar to that of the similar result for repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...s. Example Let us compute the 5-adic expansion of \frac 13. BĂ©zout's identity for 5 and the denominator 3 is 2\cdot 3 + (-1)\cdot 5 =1 (for larger examples, this can be computed with the extended Euclidean algorithm In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of BĂ©zout's ide ...). Thus :\frac 13= 2-\frac 53. For the next step, one has to "divide" -1/3 (the factor 5 in the numerator of the fraction has to be viewed as a " shift" of the -adic valuation, and thus it is not involved in the "division"). Multiplying BĂ©zout's identity by -1 gives :-\frac 13=-2+\frac 53. The "integer part" -2 is not in the right interval. So, one has to use Euclidean division by 5 for getting -2= 3-1\cdot 5, giving :-\frac 13=3-5+\frac 53 = 3-\frac 3, and :\frac 13= 2+3\cdot 5 + \frac 3\cdot 5^2. Similarly, one has :-\frac 23=1-\frac 53, and :\frac 13=2+3\cdot 5 + 1\cdot 5^2 +\frac 3\cdot 5^3. As the "remainder" -\tfrac 13 has already been found, the process can be continued easily, giving coefficients 3 for odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ... powers of five, and 1 for even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ... powers. Or in the standard 5-adic notation :\frac 13= \ldots 1313132_5 with the ellipsis The ellipsis (, also known informally as dot dot dot) is a series of dots that indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning. The plural is ellipses. The term origin ... \ldots on the left hand side. ''p''-adic series In this article, given a prime number , a ''-adic series'' is a formal series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ... of the form :\sum_^\infty a_i p^i, where every nonzero a_i is a rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ... a_i=\tfrac , such that none of n_i and d_i is divisible by . Every rational number may be viewed as a -adic series with a single term, consisting of its factorization of the form p^k\tfrac nd, with and both coprime with . A -adic series is ''normalized'' if each a_i is an integer in the interval ,p-1 So, the -adic expansion of a rational number is a normalized -adic series. The -adic valuation, or -adic order of a nonzero -adic series is the lowest integer such that a_i\ne 0. The order of the zero series is infinity \infty. Two -adic series are ''equivalent'' if they have the same order , and if for every integer the difference between their partial sums :\sum_^n a_ip^i-\sum_^n b_ip^i=\sum_^n (a_i-b_i)p^i has an order greater than (that is, is a rational number of the form p^k\tfrac ab, with k>n, and and both coprime with ). For every -adic series S, there is a unique normalized series N such that S and N are equivalent. N is the ''normalization'' of S. The proof is similar to the existence proof of the -adic expansion of a rational number. In particular, every rational number can be considered as a -adic series 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 -adic series is an equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ..., and each equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ... contains exactly one normalized -adic series. The usual operations of series (addition, subtraction, multiplication, division) map -adic series to -adic series, and are compatible with equivalence of -adic series. That is, denoting the equivalence with , if , and are nonzero -adic series such that S\sim T, one has :\begin S\pm U&\sim T\pm U,\\ SU&\sim TU,\\ 1/S&\sim 1/T. \end Moreover, and have the same order, and the same first term. Positional notation It is possible to use a positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the HinduâArabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ... similar to that which is used to represent numbers in base . Let \sum_^\infty a_i p^i be a normalized -adic series, i.e. each a_i is an integer in the interval ,p-1 One can suppose that k\le 0 by setting a_i=0 for 0\le i (if k>0), and adding the resulting zero terms to the series. If k\ge 0, the positional notation consists of writing the a_i consecutively, ordered by decreasing values of , often with appearing on the right as an index: :\ldots a_n \ldots a_1_p So, the computation of the example above shows that :\frac 13= \ldots 1313132_5, and :\frac 3= \ldots 131313200_5. When k<0, a separating dot is added before the digits with negative index, and, if the index is present, it appears just after the separating dot. For example, :\frac 1= \ldots 3131313._52, and :\frac 1= \ldots 1313131._532. If a -adic representation is finite on the left (that is, a_i=0 for large values of ), then it has the value of a nonnegative rational number of the form n p^v, with n,v integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in base . For these rational numbers, the two representations are the same. Definition There are several equivalent definitions of -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 ), completion of a metric space (see ), or inverse limits (see ). A -adic number can be defined as a ''normalized -adic series''. Since there are other equivalent definitions that are commonly used, one says often that a normalized -adic series ''represents'' a -adic number, instead of saying that it ''is'' a -adic number. One can say also that any -adic series represents a -adic number, since every -adic series is equivalent to a unique normalized -adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of -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 -adic numbers, since the series operations are compatible with equivalence of -adic series. With these operations, -adic numbers form a field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ... called the field of -adic numbers and denoted \Q_p or \mathbf Q_p. There is a unique field homomorphism Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative ri ... from the rational numbers into the -adic numbers, which maps a rational number to its -adic expansion. The image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ... of this homomorphism is commonly identified with the field of rational numbers. This allows considering the -adic numbers as an extension field of the rational numbers, and the rational numbers as a subfield of the -adic numbers. The ''valuation'' of a nonzero -adic number , commonly denoted v_p(x), is the exponent of in the first nonzero term of every -adic series that represents . By convention, v_p(0)=\infty; that is, the valuation of zero is \infty. This valuation is a discrete valuation In mathematics, a discrete valuation is an integer valuation on a field ''K''; that is, a function: :\nu:K\to\mathbb Z\cup\ satisfying the conditions: :\nu(x\cdot y)=\nu(x)+\nu(y) :\nu(x+y)\geq\min\big\ :\nu(x)=\infty\iff x=0 for all x,y\in K. .... The restriction of this valuation to the rational numbers is the -adic valuation of \Q, that is, the exponent in the factorization of a rational number as \tfrac nd p^v, with both and coprime with . ''p''-adic integers The -adic integers are the -adic numbers with a nonnegative valuation. A -adic integer can be represented as a sequence : x = (x_1 \operatorname p, ~ x_2 \operatorname p^2, ~ x_3 \operatorname p^3, ~ \ldots) of residues mod for each integer , satisfying the compatibility relations x_i \equiv x_j ~ (\operatorname p^i) for . Every integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (â1, â2, â3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ... is a -adic integer (including zero, since 0<\infty). The rational numbers of the form \tfrac nd p^k with coprime with and k\ge 0 are also -adic integers (for the reason that has an inverse mod for every ). The -adic integers form a commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ..., denoted \Z_p or \mathbf Z_p, that has the following properties. * It is an integral domain, since it is a subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ... of a field, or since the first term of the series representation of the product of two non zero -adic series is the product of their first terms. * The units (invertible elements) of \Z_p are the -adic numbers of valuation zero. * It is a principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ..., such that each ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ... is generated by a power of . * It is a local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ... of Krull dimension one, since its only prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...s are the zero ideal In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ... and the ideal generated by , 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 \Z_ = \, which is the localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ... of \Z at the prime ideal p\Z. The last property provides a definition of the -adic numbers that is equivalent to the above one: the field of the -adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by . Topological properties The -adic valuation allows defining an absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ... on -adic numbers: the -adic absolute value of a nonzero -adic number is :, x, _p = p^, where v_p(x) is the -adic valuation of . The -adic absolute value of 0 is , 0, _p = 0. This is an absolute value that satisfies the strong triangle inequality In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ... since, for every and one has * , x, _p = 0 if and only if x=0; * , x, _p\cdot , y, _p = , xy, _p *, x+y, _p\le \max(, x, _p,, y, _p) \le , x, _p + , y, _p. Moreover, if , x, _p \ne , y, _p, one has , x+y, _p = \max(, x, _p,, y, _p). This makes the -adic numbers a metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ..., and even an ultrametric space In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ..., with the -adic distance defined by d_p(x,y)=, x-y, _p. As a metric space, the -adic numbers form the completion of the rational numbers equipped with the -adic absolute value. This provides another way for defining the -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 -adic series, and thus a unique normalized -adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized -adic series instead of equivalence classes of Cauchy sequences). As the metric is defined from a discrete valuation, every open ball is also closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, .... More precisely, the open ball B_r(x) =\ equals the closed ball B_ =\, where is the least integer such that p^< r. Similarly, B_r = B_(x), where is the greatest integer such that p^>r. This implies that the -adic numbers form a locally compact space, and the -adic integersâthat is, the ball B_1 B_p(0)âform a compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i .... Modular properties The quotient ring \Z_p/p^n\Z_p may be identified with the ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ... \Z/p^n\Z of the integers modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ... p^n. This can be shown by remarking that every -adic integer, represented by its normalized -adic series, is congruent modulo p^n with its partial sum \sum_^a_ip^i, whose value is an integer in the interval ,p^n-1 A straightforward verification shows that this defines a ring isomorphism from \Z_p/p^n\Z_p to \Z/p^n\Z. The inverse limit of the rings \Z_p/p^n\Z_p is defined as the ring formed by the sequences a_0, a_1, \ldots such that a_i \in \Z/p^i \Z and a_i \equiv a_ \pmod for every . The mapping that maps a normalized -adic series to the sequence of its partial sums is a ring isomorphism from \Z_p to the inverse limit of the \Z_p/p^n\Z_p. This provides another way for defining -adic integers (up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ... an isomorphism). This definition of -adic integers is specially useful for practical computations, as allowing building -adic integers by successive approximations. For example, for computing the -adic (multiplicative) inverse of an integer, one can use Newton's method In numerical analysis, Newton's method, also known as the NewtonâRaphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ..., starting from the inverse modulo ; then, each Newton step computes the inverse modulo p^ from the inverse modulo p^n. The same method can be used for computing the -adic square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or âŻ⋅âŻ) is . For example, 4 and â4 are square roots of 16, because . E ... of an integer that is a quadratic residue In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv q \pmod. Otherwise, ''q'' is called a quadratic no ... modulo . 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 \Z_p/p^n\Z_p. Applying Newton's method to find the square root requires p^n to be larger than twice the given integer, which is quickly satisfied. Hensel lifting is a similar method that allows to "lift" the factorization modulo of a polynomial with integer coefficients to a factorization modulo p^n for large values of . This is commonly used by polynomial factorization In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field (mathematics), field or in the integers as the product of irreducible polynomial, irreducible ... algorithms. Notation There are several different conventions for writing -adic expansions. So far this article has used a notation for -adic expansions in which powers Powers may refer to: Arts and media * ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming ** ''Powers'' (American TV series), a 2015â2016 series based on the comics * ''Powers'' (British TV series), a 200 ... of increase from right to left. With this right-to-left notation the 3-adic expansion of , for example, is written as :\dfrac=\dots 121012102_3. When performing arithmetic in this notation, digits are carried to the left. It is also possible to write -adic expansions so that the powers of increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of is :\dfrac=2.01210121\dots_3\mbox\dfrac=20.1210121\dots_3. -adic expansions may be written with other sets of digits instead of . For example, the 3-adic expansion of 1/5 can be written using balanced ternary Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values â1, 0, and 1. This stands in contrast to the standard (unbalance ... digits as :\dfrac=\dots\underline11\underline11\underline11\underline_ . In fact any set of integers which are in distinct residue classes modulo may be used as -adic digits. In number theory, TeichmĂŒller representatives are sometimes used as digits. is a variant of the -adic representation of rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...s that was proposed in 1979 by Eric Hehner The given name Eric, Erich, Erikk, Erik, Erick, or Eirik is derived from the Old Norse name ''EirĂkr'' (or ''ErĂkr'' in Old East Norse due to monophthongization). The first element, ''ei-'' may be derived from the older Proto-Norse ''* ain ... and Nigel Horspool R. Nigel Horspool is a retired professor of computer science, formerly of the University of Victoria. He invented the BoyerâMooreâHorspool algorithm, a fast string search algorithm adapted from the BoyerâMoore string-search algorithm. Horsp ... for implementing on computers the (exact) arithmetic with these numbers. Cardinality Both \Z_p and \Q_p are uncountable and have the cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb .... For \Z_p, this results from the -adic representation, which defines a bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ... of \Z_p on the power set \^\N. For \Q_p this results from its expression as a countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ... union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ... of copies of \Z_p: :\Q_p=\bigcup_^\infty \frac 1\Z_p. Algebraic closure contains and is a field of characteristic . Because can be written as sum of squares, cannot be turned into an ordered field. has only a single proper algebraic extension: ; in other words, this quadratic extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ' ... is already algebraically closed. By contrast, the algebraic closure of , denoted \overline, has infinite degree, that is, has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the -adic valuation to \overline, the latter is not (metrically) complete. Its (metric) completion is called or . Here an end is reached, as is algebraically closed. However unlike this field is not locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev .... and are isomorphic as rings, so we may regard as endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ..., and does not provide an explicit example of such an isomorphism (that is, it is not constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...). If is a finite Galois extension of , the Galois group \operatorname \left(\mathbf/ \mathbf_p \right) is solvable. Thus, the Galois group \operatorname \left(\overline/ \mathbf_p \right) is prosolvable In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a .... Multiplicative group contains the -th cyclotomic field () if and only if . For instance, the -th cyclotomic field is a subfield of if and only if , or . In particular, there is no multiplicative - torsion in , if . Also, is the only non-trivial torsion element in . Given a natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ... , the index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ... of the multiplicative group of the -th powers of the non-zero elements of in \mathbf_p^ is finite. The number , defined as the sum of reciprocals of factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...s, is not a member of any -adic field; but . For one must take at least the fourth power. (Thus a number with similar properties as â namely a -th root of â is a member of \overline for all .) Localâglobal principle Helmut Hasse Helmut Hasse (; 25 August 1898 â 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...'s localâglobal principle is said to hold for an equation if it can be solved over the rational numbers if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ... it can be solved over the real numbers and over the -adic numbers for every prime . This principle holds, for example, for equations given by quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...s, but fails for higher polynomials in several indeterminates. Rational arithmetic with Hensel lifting Generalizations and related concepts The reals and the -adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...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 In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ... ''P'' of ''D''. If ''x'' is a non-zero element of ''E'', then ''xD'' is a fractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of ''D''. We write ord''P''(''x'') for the exponent of ''P'' in this factorization, and for any choice of number ''c'' greater than 1 we can set :, x, _P = c^. 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 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 , . , ''P''. 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 In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...), which are seen as encoding "local" information. This is accomplished by adele rings and idele group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; t ...s. ''p''-adic integers can be extended to ''p''-adic solenoids \mathbb_p. There is a map from \mathbb_p to the circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \ ... whose fibers are the ''p''-adic integers \mathbb_p, in analogy to how there is a map from \mathbb to the circle whose fibers are \mathbb. See also * Non-archimedean * p-adic quantum mechanics ''p''-adic quantum mechanics is a collection of related research efforts in quantum physics that replace real numbers with ''p''-adic numbers. Historically, this research was inspired by the discovery that the Veneziano amplitude of the open boson ... * p-adic Hodge theory * p-adic Teichmuller theory In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ... * p-adic analysis In mathematics, ''p''-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers. The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of lo ... * 1 + 2 + 4 + 8 + ... * ''k''-adic notation * C-minimal theory * Hensel's lemma * Locally compact field In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space.. These kinds of fields were originally introduced in p-adic analysis since the fields \mathbb_p are locally compact topological spac ... * Mahler's theorem In mathematics, Mahler's theorem, introduced by , expresses continuous ''p''-adic functions in terms of polynomials. Over any field of characteristic 0, one has the following result: Let (\Delta f)(x)=f(x+1)-f(x) be the forward difference operat ... * Profinite integer In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p where :\varprojlim \mathbb/n\mathbb indicates the profinite completion of \mathbb ... * Volkenborn integral In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions. Definition Let :f:\Z_p\to \Complex_p be a function from the p-adic integers taking values in the p-adic numbers. The Vo ... Footnotes Notes Citations References * *. — Translation into English by John Stillwell of ''Theorie der algebraischen Functionen einer VerĂ€nderlichen'' (1882). * * * * * * * * Further reading * * * * * External links *''p''-adic numberat Springer On-line Encyclopaedia of MathematicsCompletion of Algebraic Closureâ on-line lecture notes by Brian ConradAn Introduction to ''p''-adic Numbers and ''p''-adic Analysis- on-line lecture notes by Andrew Baker, 2007Efficient p-adic arithmetic(slides)Introduction to p-adic numbers* {{DEFAULTSORT:P-Adic Number Field (mathematics) Number theory
the following must be true: 0\le a_1 and a_2 Thus, one gets -p < a_1-a_2 < p, and since p divides a_1-a_2 it must be that a_1=a_2. The -adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a convergent series In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted :S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k. The th partial sum ... with the -adic absolute value. In the standard -adic notation, the digits are written in the same order as in a standard base- 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 -adic expansion of a rational number is eventually periodic. Conversely, a series \sum_^\infty a_i p^i, with 0\le a_i converges (for the -adic absolute value) to a rational number if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ... it is eventually periodic; in this case, the series is the -adic expansion of that rational number. The proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ... is similar to that of the similar result for repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...s. Example Let us compute the 5-adic expansion of \frac 13. BĂ©zout's identity for 5 and the denominator 3 is 2\cdot 3 + (-1)\cdot 5 =1 (for larger examples, this can be computed with the extended Euclidean algorithm In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of BĂ©zout's ide ...). Thus :\frac 13= 2-\frac 53. For the next step, one has to "divide" -1/3 (the factor 5 in the numerator of the fraction has to be viewed as a " shift" of the -adic valuation, and thus it is not involved in the "division"). Multiplying BĂ©zout's identity by -1 gives :-\frac 13=-2+\frac 53. The "integer part" -2 is not in the right interval. So, one has to use Euclidean division by 5 for getting -2= 3-1\cdot 5, giving :-\frac 13=3-5+\frac 53 = 3-\frac 3, and :\frac 13= 2+3\cdot 5 + \frac 3\cdot 5^2. Similarly, one has :-\frac 23=1-\frac 53, and :\frac 13=2+3\cdot 5 + 1\cdot 5^2 +\frac 3\cdot 5^3. As the "remainder" -\tfrac 13 has already been found, the process can be continued easily, giving coefficients 3 for odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ... powers of five, and 1 for even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ... powers. Or in the standard 5-adic notation :\frac 13= \ldots 1313132_5 with the ellipsis The ellipsis (, also known informally as dot dot dot) is a series of dots that indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning. The plural is ellipses. The term origin ... \ldots on the left hand side. ''p''-adic series In this article, given a prime number , a ''-adic series'' is a formal series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ... of the form :\sum_^\infty a_i p^i, where every nonzero a_i is a rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ... a_i=\tfrac , such that none of n_i and d_i is divisible by . Every rational number may be viewed as a -adic series with a single term, consisting of its factorization of the form p^k\tfrac nd, with and both coprime with . A -adic series is ''normalized'' if each a_i is an integer in the interval ,p-1 So, the -adic expansion of a rational number is a normalized -adic series. The -adic valuation, or -adic order of a nonzero -adic series is the lowest integer such that a_i\ne 0. The order of the zero series is infinity \infty. Two -adic series are ''equivalent'' if they have the same order , and if for every integer the difference between their partial sums :\sum_^n a_ip^i-\sum_^n b_ip^i=\sum_^n (a_i-b_i)p^i has an order greater than (that is, is a rational number of the form p^k\tfrac ab, with k>n, and and both coprime with ). For every -adic series S, there is a unique normalized series N such that S and N are equivalent. N is the ''normalization'' of S. The proof is similar to the existence proof of the -adic expansion of a rational number. In particular, every rational number can be considered as a -adic series 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 -adic series is an equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ..., and each equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ... contains exactly one normalized -adic series. The usual operations of series (addition, subtraction, multiplication, division) map -adic series to -adic series, and are compatible with equivalence of -adic series. That is, denoting the equivalence with , if , and are nonzero -adic series such that S\sim T, one has :\begin S\pm U&\sim T\pm U,\\ SU&\sim TU,\\ 1/S&\sim 1/T. \end Moreover, and have the same order, and the same first term. Positional notation It is possible to use a positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the HinduâArabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ... similar to that which is used to represent numbers in base . Let \sum_^\infty a_i p^i be a normalized -adic series, i.e. each a_i is an integer in the interval ,p-1 One can suppose that k\le 0 by setting a_i=0 for 0\le i (if k>0), and adding the resulting zero terms to the series. If k\ge 0, the positional notation consists of writing the a_i consecutively, ordered by decreasing values of , often with appearing on the right as an index: :\ldots a_n \ldots a_1_p So, the computation of the example above shows that :\frac 13= \ldots 1313132_5, and :\frac 3= \ldots 131313200_5. When k<0, a separating dot is added before the digits with negative index, and, if the index is present, it appears just after the separating dot. For example, :\frac 1= \ldots 3131313._52, and :\frac 1= \ldots 1313131._532. If a -adic representation is finite on the left (that is, a_i=0 for large values of ), then it has the value of a nonnegative rational number of the form n p^v, with n,v integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in base . For these rational numbers, the two representations are the same. Definition There are several equivalent definitions of -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 ), completion of a metric space (see ), or inverse limits (see ). A -adic number can be defined as a ''normalized -adic series''. Since there are other equivalent definitions that are commonly used, one says often that a normalized -adic series ''represents'' a -adic number, instead of saying that it ''is'' a -adic number. One can say also that any -adic series represents a -adic number, since every -adic series is equivalent to a unique normalized -adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of -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 -adic numbers, since the series operations are compatible with equivalence of -adic series. With these operations, -adic numbers form a field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ... called the field of -adic numbers and denoted \Q_p or \mathbf Q_p. There is a unique field homomorphism Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative ri ... from the rational numbers into the -adic numbers, which maps a rational number to its -adic expansion. The image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ... of this homomorphism is commonly identified with the field of rational numbers. This allows considering the -adic numbers as an extension field of the rational numbers, and the rational numbers as a subfield of the -adic numbers. The ''valuation'' of a nonzero -adic number , commonly denoted v_p(x), is the exponent of in the first nonzero term of every -adic series that represents . By convention, v_p(0)=\infty; that is, the valuation of zero is \infty. This valuation is a discrete valuation In mathematics, a discrete valuation is an integer valuation on a field ''K''; that is, a function: :\nu:K\to\mathbb Z\cup\ satisfying the conditions: :\nu(x\cdot y)=\nu(x)+\nu(y) :\nu(x+y)\geq\min\big\ :\nu(x)=\infty\iff x=0 for all x,y\in K. .... The restriction of this valuation to the rational numbers is the -adic valuation of \Q, that is, the exponent in the factorization of a rational number as \tfrac nd p^v, with both and coprime with . ''p''-adic integers The -adic integers are the -adic numbers with a nonnegative valuation. A -adic integer can be represented as a sequence : x = (x_1 \operatorname p, ~ x_2 \operatorname p^2, ~ x_3 \operatorname p^3, ~ \ldots) of residues mod for each integer , satisfying the compatibility relations x_i \equiv x_j ~ (\operatorname p^i) for . Every integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (â1, â2, â3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ... is a -adic integer (including zero, since 0<\infty). The rational numbers of the form \tfrac nd p^k with coprime with and k\ge 0 are also -adic integers (for the reason that has an inverse mod for every ). The -adic integers form a commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ..., denoted \Z_p or \mathbf Z_p, that has the following properties. * It is an integral domain, since it is a subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ... of a field, or since the first term of the series representation of the product of two non zero -adic series is the product of their first terms. * The units (invertible elements) of \Z_p are the -adic numbers of valuation zero. * It is a principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ..., such that each ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ... is generated by a power of . * It is a local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ... of Krull dimension one, since its only prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...s are the zero ideal In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ... and the ideal generated by , 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 \Z_ = \, which is the localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ... of \Z at the prime ideal p\Z. The last property provides a definition of the -adic numbers that is equivalent to the above one: the field of the -adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by . Topological properties The -adic valuation allows defining an absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ... on -adic numbers: the -adic absolute value of a nonzero -adic number is :, x, _p = p^, where v_p(x) is the -adic valuation of . The -adic absolute value of 0 is , 0, _p = 0. This is an absolute value that satisfies the strong triangle inequality In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ... since, for every and one has * , x, _p = 0 if and only if x=0; * , x, _p\cdot , y, _p = , xy, _p *, x+y, _p\le \max(, x, _p,, y, _p) \le , x, _p + , y, _p. Moreover, if , x, _p \ne , y, _p, one has , x+y, _p = \max(, x, _p,, y, _p). This makes the -adic numbers a metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ..., and even an ultrametric space In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ..., with the -adic distance defined by d_p(x,y)=, x-y, _p. As a metric space, the -adic numbers form the completion of the rational numbers equipped with the -adic absolute value. This provides another way for defining the -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 -adic series, and thus a unique normalized -adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized -adic series instead of equivalence classes of Cauchy sequences). As the metric is defined from a discrete valuation, every open ball is also closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, .... More precisely, the open ball B_r(x) =\ equals the closed ball B_ =\, where is the least integer such that p^< r. Similarly, B_r = B_(x), where is the greatest integer such that p^>r. This implies that the -adic numbers form a locally compact space, and the -adic integersâthat is, the ball B_1 B_p(0)âform a compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i .... Modular properties The quotient ring \Z_p/p^n\Z_p may be identified with the ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ... \Z/p^n\Z of the integers modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ... p^n. This can be shown by remarking that every -adic integer, represented by its normalized -adic series, is congruent modulo p^n with its partial sum \sum_^a_ip^i, whose value is an integer in the interval ,p^n-1 A straightforward verification shows that this defines a ring isomorphism from \Z_p/p^n\Z_p to \Z/p^n\Z. The inverse limit of the rings \Z_p/p^n\Z_p is defined as the ring formed by the sequences a_0, a_1, \ldots such that a_i \in \Z/p^i \Z and a_i \equiv a_ \pmod for every . The mapping that maps a normalized -adic series to the sequence of its partial sums is a ring isomorphism from \Z_p to the inverse limit of the \Z_p/p^n\Z_p. This provides another way for defining -adic integers (up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ... an isomorphism). This definition of -adic integers is specially useful for practical computations, as allowing building -adic integers by successive approximations. For example, for computing the -adic (multiplicative) inverse of an integer, one can use Newton's method In numerical analysis, Newton's method, also known as the NewtonâRaphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ..., starting from the inverse modulo ; then, each Newton step computes the inverse modulo p^ from the inverse modulo p^n. The same method can be used for computing the -adic square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or âŻ⋅âŻ) is . For example, 4 and â4 are square roots of 16, because . E ... of an integer that is a quadratic residue In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv q \pmod. Otherwise, ''q'' is called a quadratic no ... modulo . 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 \Z_p/p^n\Z_p. Applying Newton's method to find the square root requires p^n to be larger than twice the given integer, which is quickly satisfied. Hensel lifting is a similar method that allows to "lift" the factorization modulo of a polynomial with integer coefficients to a factorization modulo p^n for large values of . This is commonly used by polynomial factorization In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field (mathematics), field or in the integers as the product of irreducible polynomial, irreducible ... algorithms. Notation There are several different conventions for writing -adic expansions. So far this article has used a notation for -adic expansions in which powers Powers may refer to: Arts and media * ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming ** ''Powers'' (American TV series), a 2015â2016 series based on the comics * ''Powers'' (British TV series), a 200 ... of increase from right to left. With this right-to-left notation the 3-adic expansion of , for example, is written as :\dfrac=\dots 121012102_3. When performing arithmetic in this notation, digits are carried to the left. It is also possible to write -adic expansions so that the powers of increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of is :\dfrac=2.01210121\dots_3\mbox\dfrac=20.1210121\dots_3. -adic expansions may be written with other sets of digits instead of . For example, the 3-adic expansion of 1/5 can be written using balanced ternary Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values â1, 0, and 1. This stands in contrast to the standard (unbalance ... digits as :\dfrac=\dots\underline11\underline11\underline11\underline_ . In fact any set of integers which are in distinct residue classes modulo may be used as -adic digits. In number theory, TeichmĂŒller representatives are sometimes used as digits. is a variant of the -adic representation of rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...s that was proposed in 1979 by Eric Hehner The given name Eric, Erich, Erikk, Erik, Erick, or Eirik is derived from the Old Norse name ''EirĂkr'' (or ''ErĂkr'' in Old East Norse due to monophthongization). The first element, ''ei-'' may be derived from the older Proto-Norse ''* ain ... and Nigel Horspool R. Nigel Horspool is a retired professor of computer science, formerly of the University of Victoria. He invented the BoyerâMooreâHorspool algorithm, a fast string search algorithm adapted from the BoyerâMoore string-search algorithm. Horsp ... for implementing on computers the (exact) arithmetic with these numbers. Cardinality Both \Z_p and \Q_p are uncountable and have the cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb .... For \Z_p, this results from the -adic representation, which defines a bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ... of \Z_p on the power set \^\N. For \Q_p this results from its expression as a countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ... union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ... of copies of \Z_p: :\Q_p=\bigcup_^\infty \frac 1\Z_p. Algebraic closure contains and is a field of characteristic . Because can be written as sum of squares, cannot be turned into an ordered field. has only a single proper algebraic extension: ; in other words, this quadratic extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ' ... is already algebraically closed. By contrast, the algebraic closure of , denoted \overline, has infinite degree, that is, has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the -adic valuation to \overline, the latter is not (metrically) complete. Its (metric) completion is called or . Here an end is reached, as is algebraically closed. However unlike this field is not locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev .... and are isomorphic as rings, so we may regard as endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ..., and does not provide an explicit example of such an isomorphism (that is, it is not constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...). If is a finite Galois extension of , the Galois group \operatorname \left(\mathbf/ \mathbf_p \right) is solvable. Thus, the Galois group \operatorname \left(\overline/ \mathbf_p \right) is prosolvable In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a .... Multiplicative group contains the -th cyclotomic field () if and only if . For instance, the -th cyclotomic field is a subfield of if and only if , or . In particular, there is no multiplicative - torsion in , if . Also, is the only non-trivial torsion element in . Given a natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ... , the index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ... of the multiplicative group of the -th powers of the non-zero elements of in \mathbf_p^ is finite. The number , defined as the sum of reciprocals of factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...s, is not a member of any -adic field; but . For one must take at least the fourth power. (Thus a number with similar properties as â namely a -th root of â is a member of \overline for all .) Localâglobal principle Helmut Hasse Helmut Hasse (; 25 August 1898 â 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...'s localâglobal principle is said to hold for an equation if it can be solved over the rational numbers if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ... it can be solved over the real numbers and over the -adic numbers for every prime . This principle holds, for example, for equations given by quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...s, but fails for higher polynomials in several indeterminates. Rational arithmetic with Hensel lifting Generalizations and related concepts The reals and the -adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...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 In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ... ''P'' of ''D''. If ''x'' is a non-zero element of ''E'', then ''xD'' is a fractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of ''D''. We write ord''P''(''x'') for the exponent of ''P'' in this factorization, and for any choice of number ''c'' greater than 1 we can set :, x, _P = c^. 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 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 , . , ''P''. 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 In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...), which are seen as encoding "local" information. This is accomplished by adele rings and idele group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; t ...s. ''p''-adic integers can be extended to ''p''-adic solenoids \mathbb_p. There is a map from \mathbb_p to the circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \ ... whose fibers are the ''p''-adic integers \mathbb_p, in analogy to how there is a map from \mathbb to the circle whose fibers are \mathbb. See also * Non-archimedean * p-adic quantum mechanics ''p''-adic quantum mechanics is a collection of related research efforts in quantum physics that replace real numbers with ''p''-adic numbers. Historically, this research was inspired by the discovery that the Veneziano amplitude of the open boson ... * p-adic Hodge theory * p-adic Teichmuller theory In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ... * p-adic analysis In mathematics, ''p''-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers. The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of lo ... * 1 + 2 + 4 + 8 + ... * ''k''-adic notation * C-minimal theory * Hensel's lemma * Locally compact field In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space.. These kinds of fields were originally introduced in p-adic analysis since the fields \mathbb_p are locally compact topological spac ... * Mahler's theorem In mathematics, Mahler's theorem, introduced by , expresses continuous ''p''-adic functions in terms of polynomials. Over any field of characteristic 0, one has the following result: Let (\Delta f)(x)=f(x+1)-f(x) be the forward difference operat ... * Profinite integer In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p where :\varprojlim \mathbb/n\mathbb indicates the profinite completion of \mathbb ... * Volkenborn integral In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions. Definition Let :f:\Z_p\to \Complex_p be a function from the p-adic integers taking values in the p-adic numbers. The Vo ... Footnotes Notes Citations References * *. — Translation into English by John Stillwell of ''Theorie der algebraischen Functionen einer VerĂ€nderlichen'' (1882). * * * * * * * * Further reading * * * * * External links *''p''-adic numberat Springer On-line Encyclopaedia of MathematicsCompletion of Algebraic Closureâ on-line lecture notes by Brian ConradAn Introduction to ''p''-adic Numbers and ''p''-adic Analysis- on-line lecture notes by Andrew Baker, 2007Efficient p-adic arithmetic(slides)Introduction to p-adic numbers* {{DEFAULTSORT:P-Adic Number Field (mathematics) Number theory
converges (for the -adic absolute value) to a rational number
Example
''p''-adic series
Positional notation
Definition
''p''-adic integers
Topological properties
Modular properties
Notation
Cardinality
Algebraic closure
Multiplicative group
Localâglobal principle
Rational arithmetic with Hensel lifting
Generalizations and related concepts
See also
Footnotes
Notes
Citations
References
Further reading
External links