p-adic norm

p-adic norm (plural p-adic norms)

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^-ord_p(x), where ord_p(x) is the p-adic ordinal of x;[1] the same function, extended to the p-adic numbers ℚ_p (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 ℚ_p (for example, its algebraic closure).
   • 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 ${\displaystyle \operatorname {ord} _{p}a_{r}=r\lambda _{1}}$ .

   • 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.

   • 2012, Claire C. Ralph, Santiago R. Simanca, Arithmetic Differential Operators over the p-adic Integers, Cambridge University Press, page 2,

     Given a prime ${\displaystyle p}$ , we may define the p-adic norm ${\displaystyle \Vert \ \Vert _{p}}$ over the field of rational numbers ${\displaystyle \mathbb {Q} }$ .

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

     Let ${\displaystyle K}$ be a field with discrete valuation ${\displaystyle v}$ , and ${\displaystyle {\mathcal {O}}_{K}}$ be the valuation ring.

     Definition 2.3.1 A p-adic norm on vector space ${\displaystyle V}$ over ${\displaystyle K}$ is a function ${\displaystyle \alpha :V\rightarrow \mathbb {R} }$ satisfying:

     a) ${\displaystyle \alpha (x)\geq 0}$ and ${\displaystyle \alpha (x)=0}$ if and only if ${\displaystyle x=0}$ .

     b) ${\displaystyle \alpha (ax)=\vert a\vert \alpha (x)}$ for ${\displaystyle a\in K}$ and ${\displaystyle x\in V}$ .

     c) ${\displaystyle \alpha (x+y)\leq \operatorname {sup} (\alpha (x),\alpha (y))}$ for ${\displaystyle x,y\in V}$ .

     If ${\displaystyle \alpha }$ is a p-adic norm and ${\displaystyle t>0}$ , then the dilation ${\displaystyle t\alpha }$ is a p-adic norm, and we denote by ${\displaystyle N_{p}(V)}$ the set of dilation classes of p-adic norms on ${\displaystyle V}$ .

• p-adic absolute value

• p-adic number

• p-adic order on Wikipedia.Wikipedia
• Absolute value (algebra) on Wikipedia.Wikipedia
• Ostrowski's theorem on Wikipedia.Wikipedia
• p-adic Norm on Wolfram MathWorld