algebraically closed field

An algebraically closed field is a place where every polynomial equation you can write down already has a solution — there is never a need to invent new numbers to factor things. The complex numbers are the prototype: the fundamental theorem of algebra says every nonconstant polynomial with complex coefficients has a complex root. In such a field, polynomials factor completely into linear pieces, and there are no nontrivial algebraic extensions left to build.

Precisely, a field K is algebraically closed if every nonconstant polynomial in K[x] has at least one root in K. Equivalent formulations abound: every polynomial of degree n splits into n linear factors over K; the only irreducible polynomials are the linear ones; and K has no proper algebraic extension. An algebraically closed field is therefore a terminal object for algebraic extension — you cannot go any higher algebraically.

Every field K embeds in an algebraically closed field, and the smallest such, unique up to (non-canonical) isomorphism, is its algebraic closure. Algebraically closed fields are always infinite, and their structure is governed by characteristic and transcendence degree alone: two algebraically closed fields of the same characteristic and the same uncountable transcendence degree over the prime field are isomorphic. The complex numbers C are the algebraic closure of nothing smaller you usually name, but they are algebraically closed and of characteristic 0.