p-adic absolute value

Definitions

1. 1

   A norm for the rational numbers, with some prime number p as parameter, such that any…

   A norm for the rational numbers, with some prime number p as parameter, such that any rational number of the form pᵏ(a/b) — where a, b and k are integers and a, b and p are coprime — is mapped to the rational number p⁻ᵏ and 0 is mapped to 0. (Note: any nonzero rational number can be reduced to such a form.)

   • According to Ostrowski's theorem, only three kinds of norms are possible for the set of real numbers: the trivial absolute value, the real absolute value, and the p'''-adic absolute value.ᵂᴾ
   • They both gave essentially the same proof, based on the Subspace Theorem (more precisely, Schlickewei's generalisation to p'''-adic absolute values and number fields [30] of the Subspace Theorem proved by Schmidt in 1972 [41]).

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