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The p-adic number systems were first described by Kurt Hensel in 1897. For each prime 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. This is achieved by an alternative interpretation of the concept of absolute value. The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory, but their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.
More precisely, for a given prime p, the field Qp of p-adic numbers is a extension of the rational numbers. If all of the fields Qp are collectively considered, we arrive at Helmut Hasse's local-global principle, which roughly states that an equation 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. The field Qp is also given a topology derived from a metricIn mathematics, a metric space is a set (or "space") where a distance between points is defined. History Maurice Frechet introduced metric spaces in his work Sur quelques points du calcul fonctionnel Rendic. Palermo 22(1906) 1-74. Formal definition Formal, which is itself derived from an alternative valuationValuation can mean: Valuation (finance) Valuation (mathematics). on the rational numbers. This metric is complete in the sense that every Cauchy sequenceIn mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. They are named after the French mathematician Augustin Louis Cauchy. They are of interest because, given certain condi converges. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure which gives the p-adic number systems their power and utility.
If p is a fixed prime number, then any integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which st can be written as a p-adic expansion (writing the number in "base p") in the form
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 binaryThe binary or base-two numeral system is a system for representing numbers in which a radix of two is used; that is, each digit in a binary numeral may have either of two different values. Typically, the symbols 0 and 1 are used to represent binary number expansion of 35 is 1·25 + 0·24 + 0·23 + 0·22 + 1·21 + 1·20, often written in the shorthand notation 1000112.
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:
A definite meaning is given to these sums based on Cauchy sequenceIn mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. They are named after the French mathematician Augustin Louis Cauchy. They are of interest because, given certain condis using the familiar Euclidean metricThe Euclidean distance of two points x x . x and y y . y in Euclidean n space is computed as : It is the "ordinary" distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. 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.
As an alternative, if we extend the p-adic expansions by allowing infinite sums of the form
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.)
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.
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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