A -adic number is an element of the completion of the rational numbers with respect to the -adic absolute value.
Every nontrivial absolute value on is equivalent either to the usual absolute value or to one -adic absolute value.
The nontrivial non-Archimedean absolute values on a number field , up to equivalence, are indexed by the nonzero prime ideals of its ring of integers. The value associated with is a fixed exponential of the discrete valuation .
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P-adic numbers are a system of numbers used in number theory that extend the classical notion of integers and rationals to include a different form of "closeness" or convergence. The term "p-adic" refers to a prime number \( p \), and the concept is based on an alternative metric or valuation defined by \( p \).