Zp

Z_p denotes the ring of p-adic integers associated with a prime p. It is defined as the inverse limit of the system Z/p^n Z with the natural reduction maps Z/p^{n+1} Z → Z/p^n Z. Equivalently, elements of Z_p can be written as infinite series a_0 + a_1 p + a_2 p^2 + … with digits a_i in {0, …, p−1}. Z_p is a subring of the field of p-adic numbers Q_p and is the ring of integers of Q_p; its field of fractions is Q_p. As a topological ring, Z_p is compact and totally disconnected, with the p-adic topology; it is a complete, Hausdorff topological ring and a profinite ring.

Its unique maximal ideal is p Z_p, and the residue field Z_p / p Z_p is isomorphic to the finite field F_p with p elements. The unit group Z_p^× consists of those elements with a_0 ≠ 0 mod p; there is a short exact sequence 0 → 1 + p Z_p → Z_p^× → F_p^× → 0, so Z_p^× is a product of the finite cyclic group F_p^× with a pro-p subgroup 1 + p Z_p.

Z_p contains Z as a dense subring and embeds into Q_p. It is a fundamental object in number theory and p-adic analysis, underpinning local fields, Iwasawa theory, and p-adic Hodge theory.

Note: In some sources, Z_p may denote the finite field F_p or the ring Z/pZ; notation varies by context, so clarification is advised.