P-adic number

For each prime p the p- adic numbers form a field extension of the field of rational numbers; they were first described in 1897 by Kurt Hensel. These bodies are used to solve problems in number theory, often using the local-global principle of Helmut Hasse, which - broadly speaking - stating that an equation can then be solved exactly over the rational numbers, if they have the real numbers, and can be solved by all ( however that's not generally true, for the exact meaning qv). As a metric space is complete, allowing the development of a p- adic Analysis analogous to real analysis.

Motivation

If p is a fixed prime number, then each integer in a p- adic expansion of the form

Be written (it says the number is " written to the base p", see also place value system ), where the numbers are off. Thus, the 2- adic development is just about the binary representation; For example, you write:

The well-known generalization of this description to larger sets of numbers (rational and real) is the licensing of infinite sums at the lower end, that is of the following form:

These series are convergent with respect to the usual absolute value. For example, 0.13131313 ... 5, the p- adic representation of 1/3 to base 5, the integers are in this system precisely those for which holds for all. Alternatively, one could extend the sums at the other end to infinity and shape as rows

Generate, where k is an arbitrary integer. In this way we get the body of the p- adic numbers. Those p- adic numbers satisfying for all, called p -adic integers. Analogous to the ordinary p- adic expansion we can write this as a series (to the left infinite continued ) sequence of numbers:

So clearly, there is the usual p- adic development of sums that continue to the right with smaller and smaller (negative) powers of p, and the p- adic numbers have developments that continue to the left with increasing p- powers.

With these " formal Laurent series in p" can be expected as with ordinary p- adic developments of real numbers: addition from right to left with carry, multiplication by school method. Remember you just need that transfers can continue into infinity, for example, results in the addition of and the number. The missing sign is, in fact not necessary, since all inverse (there are non-negative numbers! ) Have a p- adic representation ( 1).

Furthermore, the subtraction after school method can be carried out from right to left, possibly with an infinitely often occurring " Borgen " (you try it at 05-15 = ... 444445 ).

The division, however, is in contrast to the school method carried out from right to left, thus the result is continued to the left if the division does not " add up."

A technical problem is whether these series are at all meaningful, that is, whether they converge in any sense. Two solutions to this are discussed next.

Construction

Analytical construction

The real numbers can be constructed as a completion of the rational numbers. They are regarded in this context as equivalence classes of rational Cauchy sequences. This allows us, for example, the number 1 as 1.000 ... to write, or 0.999 ... - 1.000 ... = 0.999 applies in ....

However, this depends on the definition of a Cauchy sequence of metric used, and by, uses a different metric instead of the usual Euclidean ( Archimedean ) metric that is generated by the absolute value, we obtain other completions instead of real numbers.

P- adic amount

For a fixed predetermined prime number p, we define the p- adic amount to: Any rational number x except 0 can be written in the form with a uniquely determined integer n and two natural numbers a, b, both of which are not divisible by p. We set then and. This is a non- Archimedean amount.

For example, and is thus continued:

By this definition, the amount are large powers of p " small magnitude ". With this amount to the p- adic numbers is a discrete valuation ring is defined.

P- adic metric

The p- adic metric is defined now as:

For example, the sequence in relation to the 5 -adic metric is a null sequence, whereas the sequence is bounded, but not a Cauchy sequence, because, for every n:

The completion of the metric space is the metric space of p- adic numbers. It consists of equivalence classes of Cauchy sequences, where two Cauchy sequences equivalent means if the consequence of their pointwise p- adic distances is a null sequence. In this way a complete metric space, which is obtained ( by the well-defined component-wise combinations of the Cauchy sequences equivalence classes) also is a body in which is included.

Since the This metric is an ultra metric series converge already when the summands form a null sequence. In this body, so the above-mentioned series of the form

Immediately recognizable as a convergent, if k is an integer lying in. One can show that each element can be represented as a limit of just such a series.

Algebraic construction

Here is the ring of p- adic integers is defined first, and then its quotient field.

We define as a projective limit of the rings ( see Restklassenkongruenz ): A p-adic integer then a sequence of cosets from wherein:

Each whole integer m defining a sequence and can therefore be considered as a member of.

The component-wise defined addition and multiplication are well defined, since addition and multiplication of integers with the rest of class formation are interchangeable. For any p-adic integer, the inverse number, and any number of which the first component is not 0, has an inverse, as in the case are all relatively prime, that have an inverse modulo and the result (which of the congruence projective limit has ) is then the inverse of.

Each p-adic number can be represented as a series of the above-mentioned form (1), thereby the partial sums are currently the component of the sequence. For example, one can write the 3 -adic sequence as or in the abbreviated notation as ... 0,010,223th

The ring of p- adic integers is zero divisors, so we can form the quotient body and get the body of the p- adic numbers. Each element of said body other than 0 can be represented in the form where n is an integer, and u a p-adic invertible integer ( that is, with the first component) is. This representation is unique.

Differences to the Archimedean systems

Apart from the other " counter " the convergence of p- adic metric with respect to the archimedean metric described below value system there are the following differences:

Properties

The amount of p- adic numbers is uncountable.

The body of the p-adic number, and therefore has characteristic 0, but can not be located.

The topological space of p- adic integers is a totally disconnected compact space, the space of all p- adic numbers is locally compact and totally disconnected. When both metric spaces are complete.

The prime elements of are precisely the elements associated with number p. These are precisely the elements which must be equal; This amount is the largest occurring in amount which is less than 1, the prime elements of finite extensions are divisors of p.

Is a local ring, specifically a discrete valuation ring. Its maximal ideal is generated by p ( or any other prime element ).

Real numbers have only a single real algebraic extension of the complex numbers; already formed by adjoining a square root of the extension field is algebraically closed. In contrast, the algebraic closure of an infinite extension degree. therefore has infinitely many inequivalent algebraic extensions.

The metric can be continued to a metric on the algebraic degree, but this is not then complete. The completion of the algebraic statements with respect to this metric leads to the body, which corresponds approximately with respect to its analysis of the complex numbers.

P- adic function theory

The usual definition of the function e

Converges for all x. This circle of convergence applies to all algebraic extensions and their completions, including.

This is for all; in lies. There are algebraic extensions, in which is the p th root of the fourth root of or; these roots could be regarded as p- adic analogues of the Euler number. These numbers but have to do with the real Euler number bit.

Functions from to with derivative 0 are constant. For functions of after this sentence does not apply; For example, the function

On all the derivative 0, however, is not even locally constant at 0, where the derivative is defined analogously to the real case, the limit value of the difference quotient, and the derivative 0 is

Approximation

Are elements of, then there is a sequence in such that for each (including ) the limit of in underneath. ( This statement is sometimes called proximity set or approximation theorem. )