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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 Bézout''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.