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.

Properties

The set of p-adic integers is uncountable.

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The p-adic numbers contain the rational numbers Q and form a field of characteristic 0. This field cannot be turned into an ordered field.

Related Topics:
Characteristic - Ordered field

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The topology of the set of p-adic integers is that of a Cantor set; the topology of the set of p-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity). In particular, the space of p-adic integers is compact while the space of p-adic numbers is not; it is only locally compact.

Related Topics:
Topology - Cantor set - Compact - Locally compact

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As metric spaces, both the p-adic integers and the p-adic numbers are complete.

Related Topics:
Metric space - Complete

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The real numbers have only a single proper algebraic extension, the complex numbers; in other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of the p-adic numbers has infinite degree. Furthermore, Qp has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, the algebraic closure of Qp is not (metrically) complete. Its (metric) completion is called Ωp. Here an end is reached, as Ωp is algebraically closed.

Related Topics:
Algebraic extension - Complex number - Algebraically closed - Algebraic closure

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The field Ωp is isomorphic to the field C of complex numbers, so we may regard Ωp as the complex numbers endowed with an exotic metric. It should be noted that the existence of such a field isomorphism relies on the axiom of choice, and no explicit isomorphism can be given.

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The p-adic numbers contain the nth cyclotomic field if and only if n divides p − 1. For instance, the nth cyclotomic field is a subfield of Q13 iff n = 1, 2, 3, 4, 6, or 12.

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The number e, defined as the sum of reciprocals of factorials, is not a member of any p-adic field; but ep is a p-adic number for all p except 2, for which one must take at least the fourth power. Thus e is a member of the algebraic closure of p-adic numbers for all p.

Related Topics:
E - Factorial

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Over the reals, the only functions whose derivative is zero are the constant functions. This is not true over Qp. For instance, the function

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:f: Qp → Qp, f(x) = (1/|x|p)2 for x ≠ 0, f(0) = 0,

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has zero derivative everywhere but is not even locally constant at 0.

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Given any elements r∞, r2, r3, r5, r7, ... where rp is in Qp (and Q∞ stands for R), it is possible to find a sequence (xn) in Q such that for all p (including ∞), the limit of xn in Qp is rp.

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