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Revision as of 21:23, 15 April 2024 by 1234qwer1234qwer4(talk | contribs)(→{{mvar|p}}-adic absolute value: clearly not a "norm" as used in analysis since Q/Qp aren't even real vector spaces; "norm" in algebra can be a synonym of "absolute value" (or a generalisation to modules over domains with absolute value))
In number theory, the p-adic valuation or p-adic order of an integern is the exponent of the highest power of the prime numberp that dividesn.
It is denoted .
Equivalently, is the exponent to which appears in the prime factorization of .
The p-adic valuation is a valuation and gives rise to an analogue of the usual absolute value.
Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers, the completion of the rational numbers with respect to the -adic absolute value results in the p-adic numbers.[1]
The choice of base p in the exponentiation makes no difference for most of the properties, but supports the product formula:
where the product is taken over all primes p and the usual absolute value, denoted . This follows from simply taking the prime factorization: each prime power factor contributes its reciprocal to its p-adic absolute value, and then the usual Archimedean absolute value cancels all of them.