P-adic number

The p-adic number systems were first described by Kurt Hensel in 1897. For each prime number p, the p-adic number system extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The main use of these other systems is in number theory.

Motivation

The simplest introduction to p-adic numbers is to consider 10-adic

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numbers, which are simply integers in which you allow an infinite

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number of digits to the left, for example, the number ...9999, and

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then do arithmetic with such numbers as usual. In other words, do arithmetic

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like you would with real numbers, but with digits going off to the left

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instead of to the right. The references to

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valuations and metrics given below are simply technical devices which

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justify the ordinary operations. For example, one has the computation

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:

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rac{{...9999

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top +1}} { ...000}

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which is true because there are an infinite number of carries which never

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end, so there will never be a digit "1" on the left in the result. So a first

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10-adic result is that ...999 = −1. It follows from this that negative

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integers can be represented as digit expansions in which all

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lefthand digits are eventually equal to 9. Computer scientists

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might be reminded of two's complement notation, in which negative integers

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are coded with the leftmost bit being set to 1: in the 2-adic integers, negative integers will correspond to numbers

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in which all lefthand digits are eventually equal to 1 (in general, p − 1 for p-adic

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numbers).

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One point that confuses many people is why the p in p-adic numbers is

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always prime. As seen above, it is not absolutely necessary, as things work

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well enough in base 10. (Often the term g-adic number is used when the concept is used for a fixed composite number g. for example by Kurt Mahler). However, p-adic numbers are most useful for

Related Topics:
Composite number - Kurt Mahler

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doing calculus-type computations, and it is important to always be

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able to divide, that is, one wants to be able to work in a field. The

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point is that p-adic numbers form a field if and only if p is a

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prime power, and you get the same result for a prime power as you do

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for the prime (e.g., base 16 is just shorthand for base 2). In particular, if p is not a

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prime power, then you can always find two

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nonzero p-adic numbers A and B such that AB = 0, which removes all possibility of finding their inverses. It is an interesting exercise

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to find such numbers for p = 10, for example, the following (check

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that the products are well defined over the 10-adics):

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:

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A = prod_{n = 1}^infty ,

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qquad

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B = prod_{n = 1}^infty .

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If p is a fixed prime number, then any integer can be written as a p-adic expansion (writing the number in "base p") in the form

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:pmsum_{i=0}^n a_i p^i

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where the ai are integers in {0,...,p − 1}. This is expressed by saying that the integer has been "written in base p". For example, the 2-adic or binary expansion of 35 is 1·25 + 0·24 + 0·23 + 0·22 + 1·21 + 1·20, often written in the shorthand notation 1000112.

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The familiar approach to generalizing this description to the larger domain of the rationals (and, ultimately, to the reals) is to include sums of the form:

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:pmsum_{i=-infty}^n a_i p^i

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A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...5. In this formulation, the integers are precisely those numbers which can be represented in the form where ai = 0 for all i < 0.

Related Topics:
Cauchy sequence - Absolute value

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As an alternative, if we extend the p-adic expansions by allowing infinite sums of the form

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:sum_{i=k}^{infty} a_i p^i

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where k is some (not necessarily positive) integer, we obtain the field Qp of p-adic numbers. Those p-adic numbers for which ai = 0 for all i < 0 are also called the p-adic integers. The p-adic integers form a subring of Qp, denoted Zp. (Note: Zp is often used to represent the set of integers modulo p. If each set is needed, the latter is usually written Z/pZ or Z/p. Be sure to check the notation for any text you read.)

Related Topics:
Field - Subring

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Intuitively, as opposed to p-adic expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p (as is done for the real numbers as described above), these are numbers whose p-adic expansion to the left are allowed to go on forever. For example, the p-adic expansion of 1/3 in base 5 is ...1313132, i.e. the limit of the sequence 2, 32, 132, 3132, 13132, 313132, 1313132, ... Informally, we can see that multiplying this "infinite sum" by 3 in base 5 gives ...0000001. As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 is a p-adic integer in base 5.

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The main technical problem is to define a proper notion of infinite sum which makes these expressions meaningful - this requires the introduction of the p-adic metric. Two different but equivalent solutions to this problem are presented below.

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