Algebraic Closure

Algebraic closure of a field is the smallest algebraically closed field that contains it — a field in which every non-constant polynomial has a root. The complex numbers are the algebraic closure of the real numbers, but the construction is non-trivial and requires either the axiom of choice or a more careful constructive approach to build. The existence of an algebraic closure for every field, proved by Ernst Steinitz in 1910, is one of the foundational theorems of modern algebra: it guarantees that no polynomial equation can force us outside the system if we are willing to expand the field far enough. The process of taking an algebraic closure is not unique in a strict sense — there are many isomorphic closures — but their shared structure is what matters.