{"history":[],"article":{"id":"9dbe36de-18a8-4907-a93e-66af01ecf309","title":"P-adic number","slug":"p-adic-number","content":":::summary\n:::\n\n# \nIn [number theory](https://bliptext.com/articles/number-theory), given a [prime number](https://bliptext.com/articles/prime-number) p, the **p-adic numbers** form an extension of the [rational numbers](https://bliptext.com/articles/rational-number) which is distinct from the [real numbers](https://bliptext.com/articles/real-number), though with some similar properties; **p-adic numbers** can be written in a form similar to (possibly [infinite](https://bliptext.com/articles/infinity-mathematics)) [decimals](https://bliptext.com/articles/decimal-representation), but with digits based on a [prime number](https://bliptext.com/articles/prime-number) p rather than ten, and extending to the left rather than to the right.\nFor example, comparing the expansion of the rational number \\tfrac15 in [[Ternary numeral system|base\n3\n]] vs. the\n3\n-adic expansion,\nFormally, given a prime number p, a p-adic number can be defined as a [series](https://bliptext.com/articles/series-mathematics)\nwhere k is an [integer](https://bliptext.com/articles/integer) (possibly negative), and each a_i is an [integer](https://bliptext.com/articles/integer) such that A **p-adic [integer](https://bliptext.com/articles/integer)** is a p-adic number such that k\\ge 0.\nIn general the series that represents a p-adic number is not [convergent](https://bliptext.com/articles/convergent-series) in the usual sense, but it is [convergent](https://bliptext.com/articles/convergent-series) for the [p-adic absolute value](https://bliptext.com/articles/p-adic-absolute-value) where k is the least integer i such that a_i\\ne 0 (if all a_i are zero, one has the zero p-adic number, which has\n0\nas its p-adic absolute value).\nEvery rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value.\nThis allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the [completion](https://bliptext.com/articles/completion-metric-space) of the rational numbers for the p-adic absolute value, exactly as the real numbers are the [completion](https://bliptext.com/articles/completion-metric-space) of the rational numbers for the usual absolute value.\np-adic numbers were first described by [Kurt Hensel](https://bliptext.com/articles/kurt-hensel) in 1897, though, with hindsight, some of [Ernst Kummer's](https://bliptext.com/articles/ernst-kummer) earlier work can be interpreted as implicitly using p-adic numbers.\n# [The 3-adic integers, with selected corresponding characters on their Pontryagin dual group | upload.wikimedia.org/wikipedia/commons/c/ce/3-adic///integers///with///dual///colorings.svg]\n\n# Motivation\nRoughly speaking, [modular arithmetic](https://bliptext.com/articles/modular-arithmetic) modulo a positive integer n consists of \"approximating\" every integer by the remainder of its [division](https://bliptext.com/articles/euclidean-division) by n, called its residue modulo n. The main property of [modular arithmetic](https://bliptext.com/articles/modular-arithmetic) is that the residue modulo n of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo n. If one knows that the absolute value of the result is less than n/2, this allows a computation of the result which does not involve any integer larger than n.\nFor larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the [Chinese remainder theorem](https://bliptext.com/articles/chinese-remainder-theorem) for recovering the result modulo the product of the moduli.\nAnother method discovered by [Kurt Hensel](https://bliptext.com/articles/kurt-hensel) consists of using a prime modulus p, and applying [Hensel's lemma](https://bliptext.com/articles/hensel-s-lemma) for recovering iteratively the result modulo If the process is continued infinitely, this provides eventually a result which is a p-adic number.\n\n# Basic lemmas\nThe theory of p-adic numbers is fundamentally based on the two following lemmas\nEvery nonzero rational number can be written where v, m, and n are integers and neither m nor n is divisible by p. The exponent v is uniquely determined by the rational number and is called its p-adic valuation (this definition is a particular case of a more general definition, given below).\nThe proof of the lemma results directly from the [fundamental theorem of arithmetic](https://bliptext.com/articles/fundamental-theorem-of-arithmetic).\nEvery nonzero rational number r of valuation v can be uniquely written r=ap^v+ s, where s is a rational number of valuation greater than v, and a is an integer such that 00 and\nThe p-adic numbers are essentially obtained by continuing this infinitely to produce an [infinite series](https://bliptext.com/articles/infinite-series).\n\n# p-adic series\nThe p-adic numbers are commonly defined by means of p-adic series.\nA p-adic series is a [formal power series](https://bliptext.com/articles/formal-power-series) of the form\nwhere v is an integer and the r_i are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of r_i is not divisible by p).\nEvery rational number may be viewed as a p-adic series with a single nonzero term, consisting of its factorization of the form with n and d both coprime with p.\nTwo p-adic series and\nare equivalent if there is an integer N such that, for every integer n>N, the rational number\nis zero or has a p-adic valuation greater than n.\nA p-adic series is normalized if either all a_i are integers such that and a_v >0, or all a_i are zero.\nIn the latter case, the series is called the zero series.\nEvery p-adic series is equivalent to exactly one normalized series.\nThis normalized series is obtained by a sequence of transformations, which are equivalences of series; see [§ Normalization of a p-adic series](https://bliptext.com/articles/), below.\nIn other words, the equivalence of p-adic series is an [equivalence relation](https://bliptext.com/articles/equivalence-relation), and each [equivalence class](https://bliptext.com/articles/equivalence-class) contains exactly one normalized p-adic series.\nThe usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of p-adic series.\nThat is, denoting the equivalence with\n~\n, if S, T and U are nonzero p-adic series such that S\\sim T, one has\nThe p-adic numbers are often defined as the equivalence classes of p-adic series, in a similar way as the definition of the real numbers as equivalence classes of [Cauchy sequences](https://bliptext.com/articles/cauchy-sequence).\nThe uniqueness property of normalization, allows uniquely representing any p-adic number by the corresponding normalized p-adic series.\nThe compatibility of the series equivalence leads almost immediately to basic properties of p-adic numbers:\n\n## Normalization of a p-adic series\nStarting with the series the first above lemma allows getting an equivalent series such that the p-adic valuation of r_v is zero.\nFor that, one considers the first nonzero r_i.\nIf its p-adic valuation is zero, it suffices to change v into i, that is to start the summation from v. Otherwise, the p-adic valuation of r_i is j>0, and r_i= p^js_i where the valuation of s_i is zero; so, one gets an equivalent series by changing r_i to\n0\nand r_{i+j} to Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation of r_v is zero.\nThen, if the series is not normalized, consider the first nonzero r_i that is not an integer in the interval [0,p-1].\nThe second above lemma allows writing it one gets n equivalent series by replacing r_i with a_i, and adding s_i to r_{i+1}.\nIterating this process, possibly infinitely many times, provides eventually the desired normalized\np\n-adic series.\n\n# Definition\nThere are several equivalent definitions of p-adic numbers.\nThe one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections.\nOther equivalent definitions use [completion](https://bliptext.com/articles/completion-of-a-ring) of a [discrete valuation ring](https://bliptext.com/articles/discrete-valuation-ring) (see ), [completion](https://bliptext.com/articles/completion-of-a-ring) of a metric space (see ), or [inverse limits](https://bliptext.com/articles/inverse-limit) (see ).\nA p-adic number can be defined as a normalized p-adic series.\nSince there are other equivalent definitions that are commonly used, one says often that a normalized p-adic series represents a p-adic number, instead of saying that it is a p-adic number.\nOne can say also that any p-adic series represents a p-adic number, since every p-adic series is equivalent to a unique normalized p-adic series.\nThis is useful for defining operations (addition, subtraction, multiplication, division) of p-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series.\nThis well defines operations on p-adic numbers, since the series operations are compatible with equivalence of p-adic series.\nWith these operations, p-adic numbers form a [field](https://bliptext.com/articles/field-mathematics) called the '''[field](https://bliptext.com/articles/field-mathematics) of\np\n-adic numbers''' and denoted \\Q_p or There is a unique [field homomorphism](https://bliptext.com/articles/field-homomorphism) from the rational numbers into the p-adic numbers, which maps a rational number to its p-adic expansion.\nThe [image](https://bliptext.com/articles/image-mathematics) of this homomorphism is commonly identified with the field of rational numbers.\nThis allows considering the\np\n-adic numbers as an [extension field](https://bliptext.com/articles/extension-field) of the rational numbers, and the rational numbers as a [subfield](https://bliptext.com/articles/subfield-mathematics) of the\np\n-adic numbers.\nThe valuation of a nonzero p-adic number x, commonly denoted v_p(x), is the exponent of p in the first nonzero term of every p-adic series that represents x. By convention, that is, the valuation of zero is \\infty.\nThis valuation is a [discrete valuation](https://bliptext.com/articles/discrete-valuation).\nThe restriction of this valuation to the rational numbers is the p-adic valuation of \\Q, that is, the exponent v in the factorization of a rational number as with both n and d [coprime](https://bliptext.com/articles/coprime) with p.\n\n# p-adic integers\nThe **p-adic integers** are the p-adic numbers with a nonnegative valuation.\nA p-adic integer can be represented as a sequence\nof residues xe mod pe for each integer e, satisfying the compatibility relations for i < j.\nEvery [integer](https://bliptext.com/articles/integer) is a p-adic [integer](https://bliptext.com/articles/integer) (including zero, since 0<\\infty).\nThe rational numbers of the form with d coprime with p and k\\ge 0 are also p-adic integers (for the reason that d has an inverse mod pe for every e).\nThe p-adic integers form a [commutative ring](https://bliptext.com/articles/commutative-ring), denoted \\Z_p or \\mathbf Z_p, that has the following properties.\nThe 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](https://bliptext.com/articles/field-of-fractions) of the completion of the localization of the integers at the prime ideal generated by p.\n\n# Topological properties\nThe p-adic valuation allows defining an [absolute value](https://bliptext.com/articles/absolute-value-algebra) on p-adic numbers: the p-adic [absolute value](https://bliptext.com/articles/absolute-value-algebra) of a nonzero p-adic number x is where v_p(x) is the p-adic valuation of x. The p-adic [absolute value](https://bliptext.com/articles/absolute-value-algebra) of 0 is |0|_p = 0.\nThis is an absolute value that satisfies the [strong triangle inequality](https://bliptext.com/articles/strong-triangle-inequality) since, for every x and y one has\nMoreover, if one has\nThis makes the p-adic numbers a [metric space](https://bliptext.com/articles/metric-space), and even an ultra[metric space](https://bliptext.com/articles/metric-space), with the p-adic distance defined by\nAs a metric space, the p-adic numbers form the [completion](https://bliptext.com/articles/completion-metric-space) of the rational numbers equipped with the p-adic absolute value.\nThis provides another way for defining the p-adic numbers.\nHowever, 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](https://bliptext.com/articles/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](https://bliptext.com/articles/partial-sum) of a p-adic series, and thus a unique normalized p-adic series can be associated to every equivalence class of [Cauchy sequence](https://bliptext.com/articles/cauchy-sequence)s; so, for building the completion, it suffices to consider normalized p-adic series instead of equivalence classes of [Cauchy sequence](https://bliptext.com/articles/cauchy-sequence)s).\nAs the metric is defined from a discrete valuation, every [open ball](https://bliptext.com/articles/open-ball) is also [closed](https://bliptext.com/articles/closed-ball).\nMore precisely, the open ball equals the closed ball where v is the least integer such that p^{-v}< r. Similarly, where w is the greatest integer such that p^{-w}>r.\nThis implies that the p-adic numbers form a [locally compact space](https://bliptext.com/articles/locally-compact-space), and the p-adic integers—that is, the ball —form a [compact space](https://bliptext.com/articles/compact-space).\n\n# p-adic expansion of rational numbers\nThe [decimal expansion](https://bliptext.com/articles/decimal-expansion) of a positive [rational number](https://bliptext.com/articles/rational-number) r is its representation as a [series](https://bliptext.com/articles/series-mathematics)\nwhere k is an [integer](https://bliptext.com/articles/integer) and each a_i is also an [integer](https://bliptext.com/articles/integer) such that This expansion can be computed by [long division](https://bliptext.com/articles/long-division) of the numerator by the denominator, which is itself based on the following theorem: If is a rational number such that there is an [integer](https://bliptext.com/articles/integer) a such that 0< a <10, and with r'<10^k.\nThe decimal expansion is obtained by repeatedly applying this result to the remainder r' which in the iteration assumes the role of the original rational number r.\nThe p-adic expansion of a rational number is defined similarly, but with a different division step.\nMore precisely, given a fixed [prime number](https://bliptext.com/articles/prime-number) p, every nonzero rational number r can be uniquely written as where k is a (possibly negative) integer, n and d are [coprime integers](https://bliptext.com/articles/coprime-integer) both coprime with p, and d is positive.\nThe integer k is the **p-adic valuation** of r, denoted v_p(r), and p^{-k} is its **p-adic absolute value**, denoted |r|_p (the absolute value is small when the valuation is large).\nThe division step consists of writing\nwhere a is an integer such that 0\\le a k).\nThe p-adic expansion of r is the [formal power series](https://bliptext.com/articles/formal-power-series)\nobtained by repeating indefinitely the [above](https://bliptext.com/articles/) division step on successive remainders.\nIn a p-adic expansion, all a_i are integers such that\nIf with n > 0, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation of r in [base-p](https://bliptext.com/articles/base-n).\nThe existence and the computation of the p-adic expansion of a rational number results from [Bézout's identity](https://bliptext.com/articles/b-zout-s-identity) in the following way.\nIf, as above, and d and p are coprime, there exist integers t and u such that t d+u p=1.\nSo Then, the [Euclidean division](https://bliptext.com/articles/euclidean-division) of n t by p gives\nwith 0\\le a 0), and adding the resulting zero terms to the series.\nIf k\\ge 0, the positional notation consists of writing the a_i consecutively, ordered by decreasing values of i, often with p appearing on the right as an index:\nSo, the computation of the [example above](https://bliptext.com/articles/) shows that\nand\nWhen k<0, a separating dot is added before the digits with negative index, and, if the index p is present, it appears just after the separating dot.\nFor example,\nand\nIf a p-adic representation is finite on the left (that is, a_i=0 for large values of i), then it has the value of a nonnegative rational number of the form n p^v, with n,v integers.\nThese rational numbers are exactly the nonnegative rational numbers that have a finite representation in [base](https://bliptext.com/articles/radix) p. For these rational numbers, the two representations are the same.\n\n# Modular properties\nThe [quotient ring](https://bliptext.com/articles/quotient-[ring](https://bliptext.com/articles/ring-mathematics)) may be identified with the [ring](https://bliptext.com/articles/ring-mathematics) \\Z/p^n\\Z of the integers [modulo](https://bliptext.com/articles/modular-arithmetic) p^n.\nThis can be shown by remarking that every p-adic integer, represented by its normalized p-adic series, is congruent modulo p^n with its [partial sum](https://bliptext.com/articles/partial-sum) whose value is an integer in the interval [0,p^n-1].\nA straightforward verification shows that this defines a [ring isomorphism](https://bliptext.com/articles/ring-isomorphism) from to \\Z/p^n\\Z.\nThe [inverse limit](https://bliptext.com/articles/inverse-limit) of the rings is defined as the ring formed by the sequences such that and for every i.\nThe mapping that maps a normalized p-adic series to the sequence of its partial sums is a ring isomorphism from \\Z_p to the inverse limit of the This provides another way for defining p-adic integers ([up to](https://bliptext.com/articles/up-to) an isomorphism).\nThis definition of p-adic integers is specially useful for practical computations, as allowing building p-adic integers by successive approximations.\nFor example, for computing the p-adic (multiplicative) inverse of an integer, one can use [Newton's method](https://bliptext.com/articles/newton-s-method), starting from the inverse modulo p; then, each Newton step computes the inverse modulo p^{n^2} from the inverse modulo p^n.\nThe same method can be used for computing the p-adic [square root](https://bliptext.com/articles/square-root) of an integer that is a [quadratic residue](https://bliptext.com/articles/quadratic-residue) modulo p. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found in.\nApplying Newton's method to find the square root requires p^n to be larger than twice the given integer, which is quickly satisfied.\n[Hensel lifting](https://bliptext.com/articles/hensel-lifting) is a similar method that allows to \"lift\" the factorization modulo p of a polynomial with integer coefficients to a factorization modulo p^n for large values of n. This is commonly used by [polynomial factorization](https://bliptext.com/articles/polynomial-factorization) algorithms.\n\n# Notation\nThere are several different conventions for writing p-adic expansions.\nSo far this article has used a notation for p-adic expansions in which [powers](https://bliptext.com/articles/exponentiation) of p increase from right to left.\nWith this right-to-left notation the 3-adic expansion of \\tfrac15, for example, is written as\nWhen performing arithmetic in this notation, digits are [carried](https://bliptext.com/articles/carry-arithmetic) to the left.\nIt is also possible to write p-adic expansions so that the powers of p increase from left to right, and digits are carried to the right.\nWith this left-to-right notation the 3-adic expansion of \\tfrac15 is\np-adic expansions may be written with [other sets of digits](https://bliptext.com/articles/signed-digit-representation) instead of {{math|{0, 1, ...,}} \np − 1\n}. For example, the\n3\n-adic expansion of \\tfrac15 can be written using [balanced ternary](https://bliptext.com/articles/balanced-ternary) digits\n{ 1 , 0, 1}\n, with\n1\nrepresenting negative one, as\nIn fact any set of p integers which are in distinct [residue classes](https://bliptext.com/articles/residue-class) modulo p may be used as p-adic digits.\nIn number theory, [Teichmüller representatives](https://bliptext.com/articles/witt-vector) are sometimes used as digits.\n**Quote notation** is a variant of the p-adic representation of [rational numbers](https://bliptext.com/articles/rational-number) that was proposed in 1979 by [Eric Hehner](https://bliptext.com/articles/eric-hehner) and [Nigel Horspool](https://bliptext.com/articles/nigel-horspool) for implementing on computers the (exact) arithmetic with these numbers.\n\n# Cardinality\nBoth \\Z_p and \\Q_p are [uncountable](https://bliptext.com/articles/uncountable-set) and have the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum).\nFor \\Z_p, this results from the p-adic representation, which defines a [bijection](https://bliptext.com/articles/bijection) of \\Z_p on the [power set](https://bliptext.com/articles/power-set) For \\Q_p this results from its expression as a [countably infinite](https://bliptext.com/articles/countably-infinite) [union](https://bliptext.com/articles/union-set-theory) of copies of \\Z_p:\n\n# Algebraic closure\n\\Q_p contains \\Q and is a field of [characteristic](https://bliptext.com/articles/characteristic-algebra)\n0\n.\nBecause\n0\ncan be written as sum of squares, \\Q_p cannot be turned into an [ordered field](https://bliptext.com/articles/ordered-field).\nThe field of [real numbers](https://bliptext.com/articles/real-numbers) \\R has only a single proper [algebraic extension](https://bliptext.com/articles/algebraic-extension): the [complex numbers](https://bliptext.com/articles/complex-numbers) \\C.\nIn other words, this [quadratic extension](https://bliptext.com/articles/quadratic-extension) is already [algebraically closed](https://bliptext.com/articles/algebraically-closed-field).\nBy contrast, the [algebraic closure](https://bliptext.com/articles/algebraic-closure) of \\Q_p, denoted has infinite degree, that is, \\Q_p has infinitely many inequivalent algebraic extensions.\nAlso contrasting the case of real numbers, although there is a unique extension of the p-adic valuation to the latter is not (metrically) complete.\nIts (metric) completion is called \\C_p or \\Omega_p.\nHere an end is reached, as \\C_p is algebraically closed.\nHowever unlike \\C this field is not [locally compact](https://bliptext.com/articles/locally-compact).\n\\C_p and \\C are isomorphic as rings, so we may regard \\C_p as \\C endowed with an exotic metric.\nThe proof of existence of such a field isomorphism relies on the [axiom of choice](https://bliptext.com/articles/axiom-of-choice), and does not provide an explicit example of such an isomorphism (that is, it is not [constructive](https://bliptext.com/articles/constructive-proof)).\nIf K is any finite [Galois extension](https://bliptext.com/articles/galois-extension) of \\Q_p,, the [Galois group](https://bliptext.com/articles/galois-group) is [solvable](https://bliptext.com/articles/solvable-group).\nThus, the Galois group is [prosolvable](https://bliptext.com/articles/prosolvable).\n\n# Multiplicative group\n\\Q_p contains the n-th [cyclotomic field](https://bliptext.com/articles/cyclotomic-field) (\nn > 2\n) if and only if\nn \n. For instance, the n-th cyclotomic field is a subfield of \\Q_{13} if and only if\n, or\n12\n. In particular, there is no multiplicative p-[torsion](https://bliptext.com/articles/torsion-algebra) in \\Q_p if\np > 2\n. Also,\n−1\nis the only non-trivial torsion element in \\Q_2.\nGiven a [natural number](https://bliptext.com/articles/natural-number) k, the [index](https://bliptext.com/articles/index-group-theory) of the multiplicative group of the k-th powers of the non-zero elements of \\Q_p in \\Q_p^\\times is finite.\nThe number e, defined as the sum of [reciprocals](https://bliptext.com/articles/reciprocal-mathematics) of [factorials](https://bliptext.com/articles/factorial), is not a member of any p-adic field; but for p \\ne 2.\nFor\none must take at least the fourth power.\n(Thus a number with similar properties as e — namely a p-th root of\nep\n— is a member of \\Q_p for all p.)\n\n# Local–global principle\n[Helmut Hasse](https://bliptext.com/articles/helmut-hasse)'s [local–global principle](https://bliptext.com/articles/hasse-principle) is said to hold for an equation if it can be solved over the rational numbers [if and only if](https://bliptext.com/articles/if-and-only-if) it can be solved over the real numbers and over the p-adic numbers for every prime p. This principle holds, for example, for equations given by [quadratic forms](https://bliptext.com/articles/quadratic-form), but fails for higher polynomials in several indeterminates.\n\n# Rational arithmetic with Hensel lifting\n\n# Generalizations and related concepts\nThe reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general [algebraic number fields](https://bliptext.com/articles/algebraic-number-field), in an analogous way.\nThis will be described now.\nSuppose D is a [Dedekind domain](https://bliptext.com/articles/dedekind-domain) and E is its [field of fractions](https://bliptext.com/articles/field-of-fractions).\nPick a non-zero [prime ideal](https://bliptext.com/articles/prime-ideal) P of D.\nIf x is a non-zero element of E, then xD is a [fractional ideal](https://bliptext.com/articles/fractional-ideal) and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D.\nWe write ordP(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set\nCompleting with respect to this absolute value |⋅|P yields a field EP, the proper generalization of the field of p-adic numbers to this setting.\nThe choice of c does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion).\nIt is convenient, when the [residue field](https://bliptext.com/articles/residue-field) D/P is finite, to take for c the size of D/P.\nFor example, when E is a [number field](https://bliptext.com/articles/number-field), [Ostrowski's theorem](https://bliptext.com/articles/ostrowski-s-theorem) says that every non-trivial [non-Archimedean absolute value](https://bliptext.com/articles/absolute-value-algebra) on E arises as some |⋅|P.\nThe remaining non-trivial absolute values on E arise from the different embeddings of E into the real or complex numbers.\n(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.)\nOften, one needs to simultaneously keep track of all the above-mentioned completions when E is a number field (or more generally a [global field](https://bliptext.com/articles/global-field)), which are seen as encoding \"local\" information.\nThis is accomplished by [adele rings](https://bliptext.com/articles/adele-ring) and [idele groups](https://bliptext.com/articles/idele-group).\np-adic integers can be extended to p-adic solenoids.\nThere is a map from to the [circle group](https://bliptext.com/articles/circle-group) whose fibers are the p-adic integers, in analogy to how there is a map from \\mathbb{R} to the circle whose fibers are \\mathbb{Z}.\n\n# Footnotes\n\n## Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:54:56.822Z","updated_at":"2025-01-25T17:54:56.822Z","search_vector":"'adic':3 'number':4 'p':2 'p-adic':1","search_content":" \n \n\n# \nIn [number theory](https://bliptext.com/articles/number-theory), given a [[prime number](https://bliptext.com/articles/prime-number)](https://bliptext.com/articles/prime-number) p, the **p-adic numbers** form an extension of the [rational numbers](https://bliptext.com/articles/rational-number) which is distinct from the [real numbers](https://bliptext.com/articles/real-number), though with some similar properties; **p-adic numbers** can be written in a form similar to (possibly [infinite](https://bliptext.com/articles/infinity-mathematics)) [decimals](https://bliptext.com/articles/decimal-representation), but with digits based on a [[prime number](https://bliptext.com/articles/prime-number)](https://bliptext.com/articles/prime-number) p rather than ten, and extending to the left rather than to the right.\nFor example, comparing the expansion of the rational number \\tfrac15 in [[Ternary numeral system|base\n3\n]] vs. the\n3\n-adic expansion,\nFormally, given a prime number p, a p-adic number can be defined as a [series](https://bliptext.com/articles/series-mathematics)\nwhere k is an [integer](https://bliptext.com/articles/integer) (possibly negative), and each a_i is an [integer](https://bliptext.com/articles/integer) such that A **p-adic [integer](https://bliptext.com/articles/integer)** is a p-adic number such that k\\ge 0.\nIn general the series that represents a p-adic number is not [convergent](https://bliptext.com/articles/convergent-series) in the usual sense, but it is [convergent](https://bliptext.com/articles/convergent-series) for the [p-adic absolute value](https://bliptext.com/articles/p-adic-absolute-value) where k is the least integer i such that a_i\\ne 0 (if all a_i are zero, one has the zero p-adic number, which has\n0\nas its p-adic absolute value).\nEvery rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value.\nThis allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the [completion](https://bliptext.com/articles/completion-metric-space) of the rational numbers for the p-adic absolute value, exactly as the real numbers are the [completion](https://bliptext.com/articles/completion-metric-space) of the rational numbers for the usual absolute value.\np-adic numbers were first described by [Kurt Hensel](https://bliptext.com/articles/kurt-hensel) in 1897, though, with hindsight, some of [Ernst Kummer's](https://bliptext.com/articles/ernst-kummer) earlier work can be interpreted as implicitly using p-adic numbers.\n# [The 3-adic integers, with selected corresponding characters on their Pontryagin dual group | upload.wikimedia.org/wikipedia/commons/c/ce/3-adic///integers///with///dual///colorings.svg]\n\n# Motivation\nRoughly speaking, [modular arithmetic](https://bliptext.com/articles/modular-arithmetic) modulo a positive integer n consists of \"approximating\" every integer by the remainder of its [division](https://bliptext.com/articles/euclidean-division) by n, called its residue modulo n. The main property of [modular arithmetic](https://bliptext.com/articles/modular-arithmetic) is that the residue modulo n of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo n. If one knows that the absolute value of the result is less than n/2, this allows a computation of the result which does not involve any integer larger than n.\nFor larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the [Chinese remainder theorem](https://bliptext.com/articles/chinese-remainder-theorem) for recovering the result modulo the product of the moduli.\nAnother method discovered by [Kurt Hensel](https://bliptext.com/articles/kurt-hensel) consists of using a prime modulus p, and applying [Hensel's lemma](https://bliptext.com/articles/hensel-s-lemma) for recovering iteratively the result modulo If the process is continued infinitely, this provides eventually a result which is a p-adic number.\n\n# Basic lemmas\nThe theory of p-adic numbers is fundamentally based on the two following lemmas\nEvery nonzero rational number can be written where v, m, and n are integers and neither m nor n is divisible by p. The exponent v is uniquely determined by the rational number and is called its p-adic valuation (this definition is a particular case of a more general definition, given below).\nThe proof of the lemma results directly from the [fundamental theorem of arithmetic](https://bliptext.com/articles/fundamental-theorem-of-arithmetic).\nEvery nonzero rational number r of valuation v can be uniquely written r=ap^v+ s, where s is a rational number of valuation greater than v, and a is an integer such that 0 0 and\nThe p-adic numbers are essentially obtained by continuing this infinitely to produce an [infinite series](https://bliptext.com/articles/infinite-series).\n\n# p-adic series\nThe p-adic numbers are commonly defined by means of p-adic series.\nA p-adic series is a [formal power series](https://bliptext.com/articles/formal-power-series) of the form\nwhere v is an integer and the r_i are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of r_i is not divisible by p).\nEvery rational number may be viewed as a p-adic series with a single nonzero term, consisting of its factorization of the form with n and d both coprime with p.\nTwo p-adic series and\nare equivalent if there is an integer N such that, for every integer n>N, the rational number\nis zero or has a p-adic valuation greater than n.\nA p-adic series is normalized if either all a_i are integers such that and a_v >0, or all a_i are zero.\nIn the latter case, the series is called the zero series.\nEvery p-adic series is equivalent to exactly one normalized series.\nThis normalized series is obtained by a sequence of transformations, which are equivalences of series; see [§ Normalization of a p-adic series](https://bliptext.com/articles/), below.\nIn other words, the equivalence of p-adic series is an [equivalence relation](https://bliptext.com/articles/equivalence-relation), and each [equivalence class](https://bliptext.com/articles/equivalence-class) contains exactly one normalized p-adic series.\nThe usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of p-adic series.\nThat is, denoting the equivalence with\n~\n, if S, T and U are nonzero p-adic series such that S\\sim T, one has\nThe p-adic numbers are often defined as the equivalence classes of p-adic series, in a similar way as the definition of the real numbers as equivalence classes of [Cauchy sequences](https://bliptext.com/articles/cauchy-sequence).\nThe uniqueness property of normalization, allows uniquely representing any p-adic number by the corresponding normalized p-adic series.\nThe compatibility of the series equivalence leads almost immediately to basic properties of p-adic numbers:\n\n## Normalization of a p-adic series\nStarting with the series the first above lemma allows getting an equivalent series such that the p-adic valuation of r_v is zero.\nFor that, one considers the first nonzero r_i.\nIf its p-adic valuation is zero, it suffices to change v into i, that is to start the summation from v. Otherwise, the p-adic valuation of r_i is j>0, and r_i= p^js_i where the valuation of s_i is zero; so, one gets an equivalent series by changing r_i to\n0\nand r_{i+j} to Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation of r_v is zero.\nThen, if the series is not normalized, consider the first nonzero r_i that is not an integer in the interval [0,p-1].\nThe second above lemma allows writing it one gets n equivalent series by replacing r_i with a_i, and adding s_i to r_{i+1}.\nIterating this process, possibly infinitely many times, provides eventually the desired normalized\np\n-adic series.\n\n# Definition\nThere are several equivalent definitions of p-adic numbers.\nThe one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections.\nOther equivalent definitions use [completion](https://bliptext.com/articles/completion-of-a-ring) of a [discrete valuation ring](https://bliptext.com/articles/discrete-valuation-ring) (see ), [completion](https://bliptext.com/articles/completion-of-a-ring) of a metric space (see ), or [inverse limits](https://bliptext.com/articles/inverse-limit) (see ).\nA p-adic number can be defined as a normalized p-adic series.\nSince there are other equivalent definitions that are commonly used, one says often that a normalized p-adic series represents a p-adic number, instead of saying that it is a p-adic number.\nOne can say also that any p-adic series represents a p-adic number, since every p-adic series is equivalent to a unique normalized p-adic series.\nThis is useful for defining operations (addition, subtraction, multiplication, division) of p-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series.\nThis well defines operations on p-adic numbers, since the series operations are compatible with equivalence of p-adic series.\nWith these operations, p-adic numbers form a [field](https://bliptext.com/articles/field-mathematics) called the '''[field](https://bliptext.com/articles/field-mathematics) of\np\n-adic numbers''' and denoted \\Q_p or There is a unique [field homomorphism](https://bliptext.com/articles/field-homomorphism) from the rational numbers into the p-adic numbers, which maps a rational number to its p-adic expansion.\nThe [image](https://bliptext.com/articles/image-mathematics) of this homomorphism is commonly identified with the field of rational numbers.\nThis allows considering the\np\n-adic numbers as an [extension field](https://bliptext.com/articles/extension-field) of the rational numbers, and the rational numbers as a [subfield](https://bliptext.com/articles/subfield-mathematics) of the\np\n-adic numbers.\nThe valuation of a nonzero p-adic number x, commonly denoted v_p(x), is the exponent of p in the first nonzero term of every p-adic series that represents x. By convention, that is, the valuation of zero is \\infty.\nThis valuation is a [discrete valuation](https://bliptext.com/articles/discrete-valuation).\nThe restriction of this valuation to the rational numbers is the p-adic valuation of \\Q, that is, the exponent v in the factorization of a rational number as with both n and d [coprime](https://bliptext.com/articles/coprime) with p.\n\n# p-adic integers\nThe **p-adic integers** are the p-adic numbers with a nonnegative valuation.\nA p-adic integer can be represented as a sequence\nof residues xe mod pe for each integer e, satisfying the compatibility relations for i r.\nThis implies that the p-adic numbers form a [locally [compact space](https://bliptext.com/articles/compact-space)](https://bliptext.com/articles/locally-compact-space), and the p-adic integers—that is, the ball —form a [compact space](https://bliptext.com/articles/compact-space).\n\n# p-adic expansion of rational numbers\nThe [decimal expansion](https://bliptext.com/articles/decimal-expansion) of a positive [rational number](https://bliptext.com/articles/rational-number) r is its representation as a [series](https://bliptext.com/articles/series-mathematics)\nwhere k is an [integer](https://bliptext.com/articles/integer) and each a_i is also an [integer](https://bliptext.com/articles/integer) such that This expansion can be computed by [long division](https://bliptext.com/articles/long-division) of the numerator by the denominator, which is itself based on the following theorem: If is a rational number such that there is an [integer](https://bliptext.com/articles/integer) a such that 0 k).\nThe p-adic expansion of r is the [formal power series](https://bliptext.com/articles/formal-power-series)\nobtained by repeating indefinitely the [above](https://bliptext.com/articles/) division step on successive remainders.\nIn a p-adic expansion, all a_i are integers such that\nIf with n > 0, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation of r in [base-p](https://bliptext.com/articles/base-n).\nThe existence and the computation of the p-adic expansion of a rational number results from [Bézout's identity](https://bliptext.com/articles/b-zout-s-identity) in the following way.\nIf, as above, and d and p are coprime, there exist integers t and u such that t d+u p=1.\nSo Then, the [Euclidean division](https://bliptext.com/articles/euclidean-division) of n t by p gives\nwith 0\\le a 0), and adding the resulting zero terms to the series.\nIf k\\ge 0, the positional notation consists of writing the a_i consecutively, ordered by decreasing values of i, often with p appearing on the right as an index:\nSo, the computation of the [example above](https://bliptext.com/articles/) shows that\nand\nWhen k 2\n) if and only if\nn \n. For instance, the n-th cyclotomic field is a subfield of \\Q_{13} if and only if\n, or\n12\n. In particular, there is no multiplicative p-[torsion](https://bliptext.com/articles/torsion-algebra) in \\Q_p if\np > 2\n. Also,\n−1\nis the only non-trivial torsion element in \\Q_2.\nGiven a [natural number](https://bliptext.com/articles/natural-number) k, the [index](https://bliptext.com/articles/index-group-theory) of the multiplicative group of the k-th powers of the non-zero elements of \\Q_p in \\Q_p^\\times is finite.\nThe number e, defined as the sum of [reciprocals](https://bliptext.com/articles/reciprocal-mathematics) of [factorials](https://bliptext.com/articles/factorial), is not a member of any p-adic field; but for p \\ne 2.\nFor\none must take at least the fourth power.\n(Thus a number with similar properties as e — namely a p-th root of\nep\n— is a member of \\Q_p for all p.)\n\n# Local–global principle\n[Helmut Hasse](https://bliptext.com/articles/helmut-hasse)'s [local–global principle](https://bliptext.com/articles/hasse-principle) is said to hold for an equation if it can be solved over the rational numbers [if and only if](https://bliptext.com/articles/if-and-only-if) it can be solved over the real numbers and over the p-adic numbers for every prime p. This principle holds, for example, for equations given by [quadratic forms](https://bliptext.com/articles/quadratic-form), but fails for higher polynomials in several indeterminates.\n\n# Rational arithmetic with Hensel lifting\n\n# Generalizations and related concepts\nThe reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general [algebraic number fields](https://bliptext.com/articles/algebraic-number-field), in an analogous way.\nThis will be described now.\nSuppose D is a [Dedekind domain](https://bliptext.com/articles/dedekind-domain) and E is its [field of fractions](https://bliptext.com/articles/field-of-fractions).\nPick a non-zero [prime ideal](https://bliptext.com/articles/prime-ideal) P of D.\nIf x is a non-zero element of E, then xD is a [fractional ideal](https://bliptext.com/articles/fractional-ideal) and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D.\nWe write ordP(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set\nCompleting with respect to this absolute value |⋅|P yields a field EP, the proper generalization of the field of p-adic numbers to this setting.\nThe choice of c does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion).\nIt is convenient, when the [residue field](https://bliptext.com/articles/residue-field) D/P is finite, to take for c the size of D/P.\nFor example, when E is a [number field](https://bliptext.com/articles/number-field), [Ostrowski's theorem](https://bliptext.com/articles/ostrowski-s-theorem) says that every non-trivial [non-Archimedean absolute value](https://bliptext.com/articles/absolute-value-algebra) on E arises as some |⋅|P.\nThe remaining non-trivial absolute values on E arise from the different embeddings of E into the real or complex numbers.\n(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.)\nOften, one needs to simultaneously keep track of all the above-mentioned completions when E is a number field (or more generally a [global field](https://bliptext.com/articles/global-field)), which are seen as encoding \"local\" information.\nThis is accomplished by [adele rings](https://bliptext.com/articles/adele-ring) and [idele groups](https://bliptext.com/articles/idele-group).\np-adic integers can be extended to p-adic solenoids.\nThere is a map from to the [circle group](https://bliptext.com/articles/circle-group) whose fibers are the p-adic integers, in analogy to how there is a map from \\mathbb{R} to the circle whose fibers are \\mathbb{Z}.\n\n# Footnotes\n\n## Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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