P-adic numbers
Overview
Completion of the rationals with respect to the p-adic metric. Hensel (1897) introduced them; they are central to number theory, algebraic geometry (e.g. Scholze's perfectoid spaces), and representation theory.
Era: 1897
Proposed at: Königsberg, Germany (now Kaliningrad)
Formal structure
Domain: x ∈ ℚ; prime p; |·|_p: ℚ → ℝ_≥0
|x|_p = p^{-v_p(x)}; ℚ_p = completion of ℚ; ℤ_p = {x : |x|_p ≤ 1}
Constraints: Ultrametric: |x+y|_p ≤ max(|x|_p, |y|_p); Ostrowski: |·|_p and |·|_∞ exhaust
Objective: Local-global (Hasse); p-adic analysis; number theory