English Dictionary

p-adic norm

noun

(number theory) A p-adic absolute value, for a given prime number p, the function, denoted |..|ₚ and defined on the rational numbers, such that |0|ₚ = 0 and, for x≠0, |x|ₚ = p^(-ordₚ(x)), where ordₚ(x) is the p-adic ordinal of x; the same function, extended to the p-adic numbers ℚₚ (the completion of the rational numbers with respect to the p-adic ultrametric defined by said absolute value); the same function, further extended to some extension of ℚₚ (for example, its algebraic closure).

The p-adic norm is a fundamental concept in number theory.

意味（1）

(number theory) A p-adic absolute value, for a given prime number p, the function, denoted |..|ₚ and defined on the rational numbers, such that |0|ₚ = 0 and, for x≠0, |x|ₚ = p^(-ordₚ(x)), where ordₚ(x) is the p-adic ordinal of x; the same function, extended to the p-adic numbers ℚₚ (the completion of the rational numbers with respect to the p-adic ultrametric defined by said absolute value); the same function, further extended to some extension of ℚₚ (for example, its algebraic closure).

By the property of the p-adic norm, (or by the “isosceles triangle principle”) we deduce that operatorname ordₚa_r=r𝜆₁.

意味（2）

(algebra) A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of p-adic absolute value).

Definition 2.3.1 A p-adic norm on vector space V over K is a function 𝛼:V→ R satisfying:

a) 𝛼(x)>0 and 𝛼(x)=0 if and only if x=0.

b) 𝛼(ax)=|a|𝛼(x) for a∈K and x∈V.

Definition 2.3.1 A p-adic norm on vector space V over K is a function 𝛼:V→ R satisfying: a) 𝛼(x)>0 and 𝛼(x)=0 if and only if x=0. b) 𝛼(ax)=|a|𝛼(x) for a∈K and x∈V. c) 𝛼(x+y)0, then the dilation t𝛼 is a p-adic norm, and we denote by N_p(V) the set of dilation classes of p-adic norms on V.