p-adic norm

English

Noun

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; 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 $\operatorname {ord} _{p}a_{r}=r\lambda _{1}$.

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

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 $K$ be a field with discrete valuation $v$, and ${\mathcal {O}}_{K}$ be the valuation ring.
     Definition 2.3.1 A p-adic norm on vector space $V$ over $K$ is a function $\alpha :V\rightarrow \mathbb {R}$ satisfying:
     a) $\alpha (x)\geq 0$ and $\alpha (x)=0$ if and only if $x=0$.
     b) $\alpha (ax)=\vert a\vert \alpha (x)$ for $a\in K$ and $x\in V$.
     c) $\alpha (x+y)\leq \operatorname {sup} (\alpha (x),\alpha (y))$ for $x,y\in V$.
     If $\alpha$ is a p-adic norm and $t>0$, then the dilation $t\alpha$ is a p-adic norm, and we denote by $N_{p}(V)$ the set of dilation classes of p-adic norms on $V$.

Synonyms

• p-adic absolute value

Related terms

• p-adic number

Further reading

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