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.

Related Topics:
Kurt Hensel - 1897 - Prime number - Number system - Arithmetic - Rational numbers - Real - Complex - Number theory

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The extension 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. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.

Related Topics:
Absolute value - Power series - ''p''-adic analysis - Calculus

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More formally, for a given prime p, the field Qp of p-adic numbers is an extension field 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 certain equations can be solved over the rational numbers if and only if they 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 metric, which is itself derived from an alternative valuation on the rational numbers. This metric is complete in the sense that every Cauchy sequence 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.

Related Topics:
Field - Extension field - Rational numbers - Helmut Hasse - Local-global principle - If and only if - Real numbers - Topology - Metric - Valuation - Complete - Cauchy sequence

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In the context of elliptic curves p-adic numbers are usually referred to as ell-adic numbers, due to the work of Jean-Pierre Serre. The prime p is often reserved for modular arithmetic of such curves.

Related Topics:
Elliptic curves - Jean-Pierre Serre - Modular arithmetic

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