p-adic norm

Noun

 * 1)  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; 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).
 * 2) * 2002, M. Ram Murty, Introduction to p-adic Analytic Number Theory,, page 114,
 * By the property of the p-adic norm, (or by the “isosceles triangle principle”) we deduce that $$\operatorname{ord}_p a_r = r\lambda_1$$.
 * 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.
 * 1)  A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of p-adic absolute value).
 * 1)  A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of p-adic absolute value).

Synonyms

 * p-adic absolute value

Related terms

 * p-adic number