p-adic norm

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;[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 ${\\displaystyle V}$.

Synonyms

• p-adic absolute value

Related terms

• p-adic number

References

1. 2011, Andrew Baker, An Introduction to p-adic Numbers and p-adic Analysis, Definition 2.5

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