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English
Noun
p-adic norm (plural p-adic norms)
- (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-ordp(x), where ordp(x) is the p-adic ordinal of x;[1] 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).
- 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 .
- 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 , we may define the p-adic norm over the field of rational numbers .
- (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 be a field with discrete valuation , and be the valuation ring.
Definition 2.3.1 A p-adic norm on vector space over is a function satisfying:
a) and if and only if .
b) for and .
c) for .
If is a p-adic norm and , then the dilation is a p-adic norm, and we denote by the set of dilation classes of p-adic norms on .
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