p-adic norms

Noun

p-adic norm (plural p-adic norms)

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

   Synonyms:

   p-adic absolute value

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).
   • 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)...2006, Kang Zuo, Representations of Fundamental Groups of Algebraic Varieties, Springer, page 20