p-adic norm

noun2

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).

   • By the property of the p-adic norm, (or by the “isosceles triangle principle”) we deduce that operatorname ordₚa_r=rλ₁. — 2002
2. •
   A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of p-adic absolute value).