Ostrowski's theorem

Ostrowski's theorem classifies nontrivial absolute values on the rational numbers: each is equivalent to the usual absolute value or to a p-adic absolute value for a prime p.

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Absolute value (mathematics): A function measuring the size of elements in a field while respecting multiplication and a triangle inequality. The theorem classifies absolute values on the rational numbers.
Archimedean property: A property of an ordered field in which the natural numbers are unbounded. Absolute values with Archimedean behavior fall into the usual class on the rationals.
Usual absolute value: The absolute value on the rational numbers that sends each number to its ordinary nonnegative magnitude. It represents the single Archimedean equivalence class in the theorem.
Completion (mathematics): A complete space or algebraic structure formed by adjoining limits of Cauchy sequences. Equivalent absolute values produce the same completion up to the natural field isomorphism.
Valuation (mathematics): A function that assigns algebraic size or divisibility data to elements of a field. Valuations provide an equivalent way to describe the absolute values in the theorem.
Non-Archimedean absolute value: An absolute value satisfying the stronger ultrametric triangle inequality. These are the alternative kind of absolute values classified by the theorem.
p-adic absolute value: An absolute value on the rationals defined from the exponent of a prime p in a number's factorization. Each prime supplies one non-Archimedean equivalence class.
Local field: A locally compact nondiscrete topological field, such as the real numbers or a p-adic field. The theorem identifies the rational completions that provide the basic characteristic-zero local fields.
Equivalent absolute values: Absolute values that induce the same topology on a field. The theorem classifies absolute values up to this equivalence, not by their exact formulas.
p-adic valuation: The exponent of a prime p in the factorization of a nonzero rational number. Its exponent data determine the absolute values in the non-Archimedean classes.