In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p rather than ten, and extending (possibly infinitely) to the left rather than to the right. Formally, given a prime number p, a p-adic number can be defined as a series
where k is an integer (possibly negative), and each is an integer such that A p-adic integer is a p-adic number such that
In general the series that represents a p-adic number is not convergent in the usual sense, but it is convergent for the p-adic absolute value where k is the least integer i such that (if all are zero, one has the zero p-adic number, which has 0 as its p-adic absolute value).
Every rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value. This allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the completion of the rational numbers for the p-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.
Roughly speaking, modular arithmetic modulo a positive integer n consists of "approximating" every integer by the remainder of its division by n, called its residue modulo n. The main property of 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.
For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the Chinese remainder theorem for recovering the result modulo the product of the moduli.
Another method discovered by Kurt Hensel consists of using a prime modulus p, and applying 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.
The theory of p-adic numbers is fundamentally based on the two following lemmas
Every 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). The proof of the lemma results directly from the fundamental theorem of arithmetic.
Every nonzero rational number r of valuation v can be uniquely written where s is a rational number of valuation greater than v, and a is an integer such that
The proof of this lemma results from modular arithmetic: By the above lemma, where m and n are integers coprime with p. The modular inverse of n is an integer q such that for some integer h. Therefore, one has and The Euclidean division of by p gives where since mq is not divisible by p. So,
which is the desired result.
This can be iterated starting from s instead of r, giving the following.
Given a nonzero rational number r of valuation v and a positive integer k, there are a rational number of nonnegative valuation and k uniquely defined nonnegative integers less than p such that and
The p-adic numbers are essentially obtained by continuing this infinitely to produce an infinite series.
The p-adic numbers are commonly defined by means of p-adic series.
A p-adic series is a formal power series of the form
where is an integer and the are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of is not divisible by p.
Every 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.
Two p-adic series and are equivalent if there is an integer N such that, for every integer the rational number
is zero or has a p-adic valuation greater than n.
A p-adic series is normalized if either all are integers such that and or all are zero. In the latter case, the series is called the zero series.
Every p-adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see § Normalization of a p-adic series, below.
The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of p-adic series. That is, denoting the equivalence with ~, if S, T and U are nonzero p-adic series such that one has
The 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. The uniqueness property of normalization, allows uniquely representing any p-adic number by the corresponding normalized p-adic series. The compatibility of the series equivalence leads almost immediately to basic properties of p-adic numbers:
- Addition, multiplication and multiplicative inverse of p-adic numbers are defined as for formal power series, followed by the normalization of the result.
- With these operations, the p-adic numbers form a field, which is an extension field of the rational numbers.
- The valuation of a nonzero p-adic number x, commonly denoted is the exponent of p in the first non zero term of the corresponding normalized series; the valuation of zero is
- The p-adic absolute value of a nonzero p-adic number x, is for the zero p-adic number, one has
Normalization of a p-adic series
Starting with the series the first above lemma allows getting an equivalent series such that the p-adic valuation of is zero. For that, one considers the first nonzero If 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 is and where the valuation of is zero; so, one gets an equivalent series by changing to 0 and 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 is zero.
Then, if the series is not normalized, consider the first nonzero that is not an integer in the interval The second above lemma allows writing it one gets n equivalent series by replacing with and adding to Iterating this process, possibly infinitely many times, provides eventually the desired normalized p-adic series.
There are several equivalent definitions of p-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use completion of a discrete valuation ring (see § p-adic integers), completion of a metric space (see § Topological properties), or inverse limits (see § Modular properties).
A p-adic number can be defined as a normalized p-adic series. Since 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.
One 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. This 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. This well defines operations on p-adic numbers, since the series operations are compatible with equivalence of p-adic series.
With these operations, p-adic numbers form a field called the field of p-adic numbers and denoted or There is a unique field homomorphism from the rational numbers into the p-adic numbers, which maps a rational number to its p-adic expansion. The image of this homomorphism is commonly identified with the field of rational numbers. This allows considering the p-adic numbers as an extension field of the rational numbers, and the rational numbers as a subfield of the p-adic numbers.
The valuation of a nonzero p-adic number x, commonly denoted 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 This valuation is a discrete valuation. The restriction of this valuation to the rational numbers is the p-adic valuation of that is, the exponent v in the factorization of a rational number as with both n and d coprime with p.
The p-adic integers are the p-adic numbers with a nonnegative valuation.
A p-adic integer can be represented as a sequence
of residues xe mod pe for each integer e, satisfying the compatibility relations for i < j.
Every integer is a p-adic integer (including zero, since ). The rational numbers of the form with d coprime with p and are also p-adic integers (for the reason that d has an inverse mod pe for every e).
The p-adic integers form a commutative ring, denoted or , that has the following properties.
- It is an integral domain, since it is a subring of a field, or since the first term of the series representation of the product of two non zero p-adic series is the product of their first terms.
- The units (invertible elements) of are the p-adic numbers of valuation zero.
- It is a principal ideal domain, such that each ideal is generated by a power of p.
- It is a local ring of Krull dimension one, since its only prime ideals are the zero ideal and the ideal generated by p, 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 which is the localization of at the prime ideal
The last property provides a definition of the p-adic numbers that is equivalent to the above one: the field of the p-adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by p.
The p-adic valuation allows defining an absolute value on p-adic numbers: the p-adic absolute value of a nonzero p-adic number x is
where is the p-adic valuation of x. The p-adic absolute value of is This is an absolute value that satisfies the strong triangle inequality since, for every x and y one has
- if and only if
Moreover, if one has
As a metric space, the p-adic numbers form the completion of the rational numbers equipped with the p-adic absolute value. This provides another way for defining the p-adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every Cauchy sequence a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the partial sums of a p-adic series, and thus a unique normalized p-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized p-adic series instead of equivalence classes of Cauchy sequences).
As the metric is defined from a discrete valuation, every open ball is also closed. More precisely, the open ball equals the closed ball where v is the least integer such that Similarly, where w is the greatest integer such that
p-adic expansion of rational numbers
where is an integer and each is also an integer such that This expansion can be computed by 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 such that and with The decimal expansion is obtained by repeatedly applying this result to the remainder which in the iteration assumes the role of the original rational number .
The p-adic expansion of a rational number is defined similarly, but with a different division step. More precisely, given a fixed prime number , every nonzero rational number can be uniquely written as where is a (possibly negative) integer, and are coprime integers both coprime with , and is positive. The integer is the p-adic valuation of , denoted and is its p-adic absolute value, denoted (the absolute value is small when the valuation is large). The division step consists of writing
where is an integer such that and is either zero, or a rational number such that (that is, ).
The -adic expansion of is the formal power series
obtained by repeating indefinitely the above division step on successive remainders. In a p-adic expansion, all are integers such that
If with , 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 in base-p.
The existence and the computation of the p-adic expansion of a rational number results from Bézout's identity in the following way. If, as above, and and are coprime, there exist integers and such that So
Then, the Euclidean division of by gives
with This gives the division step as
so that in the iteration
is the new rational number.
The uniqueness of the division step and of the whole p-adic expansion is easy: if one has This means divides Since and the following must be true: and Thus, one gets and since divides it must be that
The p-adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a convergent series with the p-adic absolute value. In the standard p-adic notation, the digits are written in the same order as in a standard base-p system, namely with the powers of the base increasing to the left. This means that the production of the digits is reversed and the limit happens on the left hand side.
The p-adic expansion of a rational number is eventually periodic. Conversely, a series with converges (for the p-adic absolute value) to a rational number if and only if it is eventually periodic; in this case, the series is the p-adic expansion of that rational number. The proof is similar to that of the similar result for repeating decimals.
Let us compute the 5-adic expansion of Bézout's identity for 5 and the denominator 3 is (for larger examples, this can be computed with the extended Euclidean algorithm). Thus
For the next step, one has to "divide" (the factor 5 in the numerator of the fraction has to be viewed as a "shift" of the p-adic valuation, and thus it is not involved in the "division"). Multiplying Bézout's identity by gives
The "integer part" is not in the right interval. So, one has to use Euclidean division by for getting giving
Similarly, one has
with the ellipsis on the left hand side.
Let be a normalized p-adic series, i.e. each is an integer in the interval One can suppose that by setting for (if ), and adding the resulting zero terms to the series.
If the positional notation consists of writing the consecutively, ordered by decreasing values of i, often with p appearing on the right as an index:
So, the computation of the example above shows that
When 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. For example,
If a p-adic representation is finite on the left (that is, for large values of i), then it has the value of a nonnegative rational number of the form with integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in base p. For these rational numbers, the two representations are the same.
The quotient ring may be identified with the ring of the integers modulo This can be shown by remarking that every p-adic integer, represented by its normalized p-adic series, is congruent modulo with its partial sum whose value is an integer in the interval A straightforward verification shows that this defines a ring isomorphism from to
The inverse limit of the rings is defined as the ring formed by the sequences such that and for every i.
The mapping that maps a normalized p-adic series to the sequence of its partial sums is a ring isomorphism from to the inverse limit of the This provides another way for defining p-adic integers (up to an isomorphism).
This definition of p-adic integers is specially useful for practical computations, as allowing building p-adic integers by successive approximations.
For example, for computing the p-adic (multiplicative) inverse of an integer, one can use Newton's method, starting from the inverse modulo p; then, each Newton step computes the inverse modulo from the inverse modulo
The same method can be used for computing the p-adic square root of an integer that is a quadratic residue modulo p. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found in . Applying Newton's method to find the square root requires 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 p of a polynomial with integer coefficients to a factorization modulo for large values of n. This is commonly used by polynomial factorization algorithms.
There are several different conventions for writing p-adic expansions. So far this article has used a notation for p-adic expansions in which powers of p increase from right to left. With this right-to-left notation the 3-adic expansion of for example, is written as
When performing arithmetic in this notation, digits are carried to the left. It 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. With this left-to-right notation the 3-adic expansion of is
Quote notation is a variant of the p-adic representation of rational numbers that was proposed in 1979 by Eric Hehner and Nigel Horspool for implementing on computers the (exact) arithmetic with these numbers.
Both and are uncountable and have the cardinality of the continuum. For this results from the p-adic representation, which defines a bijection of on the power set For this results from its expression as a countably infinite union of copies of :
Qp contains Q and is a field of characteristic 0.
R has only a single proper algebraic extension: C; in other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of Qp, denoted has infinite degree, that is, Qp has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the p-adic valuation to the latter is not (metrically) complete. Its (metric) completion is called Cp or Ωp. Here an end is reached, as Cp is algebraically closed. However unlike C this field is not locally compact.
Cp and C are isomorphic as rings, so we may regard Cp as C endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the axiom of choice, and does not provide an explicit example of such an isomorphism (that is, it is not constructive).
Qp contains the n-th cyclotomic field (n > 2) if and only if n | p − 1. For instance, the n-th cyclotomic field is a subfield of Q13 if and only if n = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicative p-torsion in Qp, if p > 2. Also, −1 is the only non-trivial torsion element in Q2.
The number e, defined as the sum of reciprocals of factorials, is not a member of any p-adic field; but e p ∈ Qp (p ≠ 2). For p = 2 one must take at least the fourth power. (Thus a number with similar properties as e — namely a p-th root of ep — is a member of for all p.)
Helmut Hasse's local–global principle is said to hold for an equation if it can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p. This principle holds, for example, for equations given by quadratic forms, but fails for higher polynomials in several indeterminates.
Rational arithmetic with Hensel lifting
The 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, in an analogous way. This will be described now.
Suppose D is a Dedekind domain and E is its field of fractions. Pick a non-zero prime ideal P of D. If x is a non-zero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D. We write ordP(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set
Completing with respect to this absolute value |⋅|P yields a field EP, 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 Cp, thus putting the description of all the non-trivial absolute values of a number field on a common footing.)
Often, one needs to simultaneously keep track of all the above-mentioned completions when E is a number field (or more generally a global field), which are seen as encoding "local" information. This is accomplished by adele rings and idele groups.
p-adic integers can be extended to p-adic solenoids . There is a map from to the circle group whose fibers are the p-adic integers , in analogy to how there is a map from to the circle whose fibers are .
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