Qp

Q_p denotes the field of p-adic numbers, the completion of the rational numbers with respect to the p-adic absolute value |·|_p. For a nonzero rational x, write x = p^{v_p(x)}(a/b) with integers a, b not divisible by p; then v_p(x) is the exponent of p in x and |x|_p = p^{-v_p(x)}. The distance is d_p(x,y) = |x - y|_p, giving a non-Archimedean (ultrametric) topology.

Q_p is obtained as the completion of Q under d_p, i.e., as equivalence classes of Cauchy sequences with respect to the p-adic metric. Each element can be represented by a convergent p-adic expansion x = sum_{i=k}^{∞} a_i p^i with digits a_i in {0,1,...,p−1} and some integer k. The ring of integers Z_p consists of elements with nonnegative valuation (k ≥ 0) and can be written as sum_{i=0}^{∞} a_i p^i. Z_p has maximal ideal pZ_p, and the residue field Z_p/pZ_p is isomorphic to the finite field F_p.

Q_p is a locally compact, complete field with the ultrametric inequality |x + y|_p ≤ max(|x|_p, |y|_p). Its multiplicative group decomposes as Q_p^× ≅ p^Z × Z_p^×, where Z_p^× is the group of units (elements with |x|_p = 1), itself structured as μ_{p−1} × (1 + pZ_p).

Applications of Q_p appear across number theory and arithmetic geometry, including local analysis, Hensel’s lemma, and the theory of local fields.