High Quality Content by WIKIPEDIA articles! Ostrowski's theorem, due to Alexander Ostrowski, states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value. Since | | is non-Archimedean, |n| ? 1 for all integers n. Also as | | is non-trivial, there exists an integer n such that |n| < 1 and n = p_1^{e_1} ldots p_r^{e_r} by integer factorization. From this, we can deduce |p| < 1 for some prime p. Suppose for contradiction p, q are distinct primes with |p|, |q| < 1. Pick e, f such that |p|e, |q|f < 1 and write 1 = rpe + sqf for some integers r, s by Bezout's identity. But then 1 = |rpe + sqf| < max(|r|, |s|) ? 1, which is a desired contradiction. So must have |p| = ?, some 0 < ? < 1, and |q| = 1 for all other primes q. Therefore | | is equivalent to the p-adic absolute value.

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