p-adic norm

English

Noun

1. Template:Lb 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, Template:W, page 114,

     By the property of the p-adic norm, (or by the “isosceles triangle principle”) we deduce that ${\mathrm{ord}}_p{a}_r=r{\lambda }_1$.

   • 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.

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

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Related terms

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Further reading

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• p-adic Norm on Template:W