Meaning of p-adic norm | Babel Free

Definitions

1. 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).
2. A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of p-adic absolute value).

Examples

“2002, M. Ram Murty, Introduction to p-adic Analytic Number Theory, American Mathematical Society, page 114, By the property of the p'''-adic norm, (or by the “isosceles triangle principle”) we deduce that operatorname ordₚa_r=rλ₁.”

“2006, Matti Pitkanen, Topological Geometrodynamics, Luniver Press, page 531, The definition of p-adic norm should obey the usual conditions, in particular the requirement that the norm of product is product of norms.”

“Given a prime p, we may define the p'''-adic norm #92;Vert#92;#92;Vert#95;p over the field of rational numbers #92;Q.”

“Let K be a field with discrete valuation v, and #92;mathcal#123;O#125;#95;K be the valuation ring. Definition 2.3.1 A p'''-adic norm on vector space V over K is a function #92;alpha#58;V#92;rightarrow#92;R satisfying: a) #92;alpha(x)#92;ge 0 and #92;alpha(x)#61;0 if and only if x#61;0. b) #92;alpha(ax)#61;#92;verta#92;vert#92;alpha(x) for a#92;inK and x#92;inV. c) #92;alpha(x#43;y)#92;le#92;operatorname#123;sup#125;(#92;alpha(x),#92;alpha(y)) for x,y#92;inV. If #92;alpha is a p'''-adic norm and tgt;0, then the dilation t#92;alpha is a p'''-adic norm, and we denote by N#95;p(V) the set of dilation classes of p'''-adic norms on V.”

CEFR level