(number theory) Of, pertaining to, (ultimately) derived from or defined in the context of p-adic numbers.

(number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.

(number theory) A function of rational numbers, with prime number p as parameter, which is defined for some non-zero integer x as the largest integer r such that pʳ divides x; is defined for some non-zero rational number a/b as the p-adic ordinal of a minus the p-adic ordinal of b; and is defined for 0 as infinity.

(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; 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).

(number theory) The ultrametric with prime number p as parameter defined as d_p(x,y)=|x-y|ₚ; i.e., such that the distance between two rational numbers is equal to the p-adic absolute value of the difference between those two numbers.

plural of p-adic norm

plural of p-adic ordinal

plural of p-adic number

plural of p-adic order

(number theory) Of a positive integer n, the exponent of the highest power of p that divides n. Often denoted νₚ(n). For example, ν₂(48)=4.