Meaning of p-adic absolute value | Babel Free

Definitions

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.)

Examples

“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”

“If cgt;1, then x#92;rightarrowc#123;-v#95;p(x)#125; (with the convention c#123;-#92;infty#125;#61;0) is a nonarchimedean absolute value, denoted #92;vert..#92;vert#95;#123;p,c#125; and called the p-adic absolute value to base c. If cgt;1 and dgt;1 and if r#61;#92;log#95;cd, then #92;vertx#92;vert#95;#123;p,d#125;#61;#92;vertx#92;vertʳ#95;#123;p,c#125; for every x#92;inK. The p-adic topology on K is the topology defined by the p'''-adic absolute values.”

“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]).”

“It ^([#92;Z#95;p]) can also be defined as the subset ^([of #92;Q#95;p]) with #92;vertx#92;vert#95;plt;p because the p'''-adic absolute value takes no values between 1 and p, and therefore #92;Z#95;p is open.”

CEFR level