Theorem

Ostrowski's Theorem

Number Theory

Ostrowski's Theorem classifies every nontrivial absolute value on the field of rational numbers as equivalent either to the ordinary real absolute value or to one of the p-adic absolute values, one for each prime number p. Proved by Alexander Ostrowski, it explains why the real numbers and the p-adic number fields are the only essentially different ways of completing the rational numbers.

Facts

Statement

Every nontrivial absolute value on the field of rational numbers is equivalent either to the usual real absolute value or to a p-adic absolute value for some prime p, and no other inequivalent absolute values on the rationals exist. Proved by Alexander Ostrowski in 1916. 1

Proof Year

1916 1

Classification

Statement Form

Classification Theorem 1

Connections

Source Ostrowski's Theorem (Wikipedia)

Sources

1. Ostrowski's Theorem (Wikipedia)

Wikimedia Foundation

lead paragraph, first sentence
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers is equivalent to either the usual real absolute value or a p-adic absolute value.
Proved By: Alexander Ostrowski, Lead section
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real

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Ostrowski's Theorem

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