P-adic number

In number theory, given a prime number, the -adic numbers form an extension of the rational numbers that is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right.
For example, comparing the expansion of the rational number in base vs. the -adic expansion,
Formally, given a prime number, a -adic number can be defined as a series
where is an integer, and each is an integer such that A -adic integer is a -adic number such that
In general the series that represents a -adic number is not convergent in the usual sense, but it is convergent for the -adic absolute value where is the least integer such that .
Every rational number can be uniquely expressed as the sum of a series as above, with respect to the -adic absolute value. This allows considering rational numbers as special -adic numbers, and alternatively defining the -adic numbers as the completion of the rational numbers for the -adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.
-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.

Motivation

Roughly speaking, modular arithmetic modulo a positive integer consists of "approximating" every integer by the remainder of its division by, called its residue modulo. The main property of modular arithmetic is that the residue modulo 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.
When studying Diophantine equations, it's sometimes useful to reduce the equation modulo a prime, since this usually provides more insight about the equation itself. Unfortunately, doing this loses some information because the reduction is not injective.
One way to preserve more information is to use larger moduli, such as higher prime powers,,. However, this has the disadvantage of not being a field, which loses a lot of the algebraic properties that has.
Kurt Hensel discovered a method which consists of using a prime modulus, and applying Hensel's lemma to lift solutions modulo to modulo,. This process creates an infinite sequence of residues, and a -adic number is defined as the "limit" of such a sequence.
Essentially, -adic numbers allows "taking modulo for all at once". A distinguishing feature of -adic numbers from ordinary modulo arithmetic is that the set of -adic numbers forms a field, making division by possible. Furthermore, the mapping is injective, so not much information is lost when reducing to -adic numbers.

Informal description

There are multiple ways to understand -adic numbers.

As a base-''p'' expansion

One way to think about -adic integers is using "base ". For example, every integer can be written in base,
Informally, -adic integers can be thought of as integers in base-, but the digits extend infinitely to the left.
Addition and multiplication on -adic integers can be carried out similarly to integers in base-.
When adding together two -adic integers, for example, their digits are added with carries being propagated from right to left.
Multiplication of -adic integers works similarly via long multiplication. Since addition and multiplication can be performed with -adic integers, they form a ring, denoted or.
Note that some rational numbers can also be -adic integers, even if they aren't integers in a real sense. For example, the rational number is a 3-adic integer, and has the 3-adic expansion. However, some rational numbers, such as, cannot be written as a -adic integer. Because of this, -adic integers are generalized further to -adic numbers:
-adic numbers can be thought of as -adic integers with finitely many digits after the decimal point. An example of a 3-adic number is
Equivalently, every -adic number is of the form, where is a -adic integer.
For any -adic number, its multiplicative inverse is also a -adic number, which can be computed using a variant of long division. For this reason, the -adic numbers form a field, denoted or.

As a sequence of residues mod ^''k''

Another way to define -adic integers is by representing it as a sequence of residues mod for each integer,, Here each denotes an integer representative of a residue class modulo
satisfying the compatibility relations for. In this notation, addition and multiplication of -adic integers are defined component-wise:
This is equivalent to the base- definition, because the last digits of a base- expansion uniquely define its value mod ^k, and vice versa.
This form can also explain why some rational numbers are -adic integers, even if they are not integers. For example, is a 3-adic integer, because its 3-adic expansion consists of the multiplicative inverses of 5 mod 3, 3², 3³,...

Definition

There are several equivalent definitions of -adic numbers. The two approaches given below are relatively elementary.

As formal series in base

A -adic integer is often defined as a formal power series of the form
where each represents a "digit in base ".
A -adic unit is a -adic integer whose first digit is nonzero, i.e.. The set of all -adic integers is usually denoted.
A -adic number is then defined as a formal Laurent series of the form
where is a integer, and each. Equivalently, a -adic number is anything of the form, where is a -adic integer.
The first index for which the digit is nonzero in is called the -adic valuation of, denoted. If, then such an index does not exist, so by convention.
In this definition, addition, subtraction, multiplication, and division of -adic numbers are carried out similarly to numbers in base, with "carries" or "borrows" moving from left to right rather than right to left. As an example in,
Division of -adic numbers may also be carried out "formally" via division of formal power series, with some care about having to "carry".
With these operations, the set of -adic numbers form a field, denoted.

As equivalence classes

The -adic numbers may also be defined as equivalence classes, in a similar way as the definition of real numbers as equivalence classes of Cauchy sequences. It is fundamentally based on the following lemma:
The exponent is uniquely determined by and is called its -adic valuation, denoted. The proof of the lemma results directly from the fundamental theorem of arithmetic.
A -adic series is a formal Laurent series of the form
where is a integer and the are rational numbers that either are zero or have a nonnegative valuation.
Every rational number may be viewed as a -adic series with a single nonzero term, consisting of its factorization of the form with and both coprime with.
Two -adic series and
are equivalent if there is an integer such that, for every integer the rational number
is zero or has a -adic valuation greater than.
A -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 -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 -adic series], below.
In other words, the equivalence of -adic series is an equivalence relation, and each equivalence class contains exactly one normalized -adic series.
The usual operations of series are compatible with equivalence of -adic series. That is, denoting the equivalence with, if, and are nonzero -adic series such that one has
With this, the -adic numbers are defined as the equivalence classes of -adic series.
The uniqueness property of normalization, allows uniquely representing any -adic number by the corresponding normalized -adic series. The compatibility of the series equivalence leads almost immediately to basic properties of -adic numbers:

Addition, multiplication and multiplicative inverse of -adic numbers are defined as for formal power series, followed by the normalization of the result.
With these operations, the -adic numbers form a field, which is an extension field of the rational numbers.
The valuation of a nonzero -adic number, commonly denoted is the exponent of in the first non zero term of the corresponding normalized series; the valuation of zero is
The -adic absolute value of a nonzero -adic number, is for the zero -adic number, one has
Normalization of a ''p''-adic series

Starting with the series we wish to arrive at an equivalent series such that the -adic valuation of is zero. For that, one considers the first nonzero If its -adic valuation is zero, it suffices to change into, that is to start the summation from. Otherwise, the -adic valuation of is and where the valuation of is zero; so, one gets an equivalent series by changing to 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 Using Bézout's lemma, write this as, where and has nonnegative valuation. Then, one gets an equivalent series by replacing with and adding to Iterating this process, possibly infinitely many times, provides eventually the desired normalized -adic series.

Other equivalent definitions

Other equivalent definitions use completion of a discrete valuation ring, completion of a metric space, or inverse limits.
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 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 called the field of -adic numbers and denoted or There is a unique field homomorphism from the rational numbers into the -adic numbers, which maps a rational number to its -adic expansion. The image 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 is the exponent of in the first nonzero term of every -adic series that represents. 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 -adic valuation of that is, the exponent in the factorization of a rational number as with both and coprime with.