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.

Generalizations and related concepts

The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way. This will be described now.

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Suppose D is a Dedekind domain and E is its quotient field. The non-zero prime ideals of D are also called finite places or finite primes of E. If x is a non-zero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of finite primes of E. If P is such a finite prime, we write ordP(x) for the exponent of P in this factorization, and define

Related Topics:
Dedekind domain - Quotient field - Prime ideal

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:|x|_P = (NP)^{-operatorname{ord}_P(x)}

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where NP denotes the (finite) cardinality of D/P. Completing with respect to this norm |.|P then yields a field EP, the proper generalization of the field of p-adic numbers to this setting.

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Often, one needs to simultaneously keep track of all the above mentioned completions, which are seen as encoding "local" information. This is accomplished by adele rings and idele groups.

Related Topics:
Adele ring - Idele group

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