algebraic closure

Some fields are leaky: you can write down a polynomial whose roots are nowhere to be found inside them. The algebraic closure of a field is the result of plugging every such leak at once — the smallest extension in which every polynomial of degree at least one finally has all its roots. Once you are inside the closure, no polynomial can escape you.

Precisely, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed, meaning every non-constant polynomial with coefficients in it factors completely into linear factors. Every field has an algebraic closure, and it is unique up to isomorphism, so we speak of the algebraic closure, often written K-bar.

The closure is built only from algebraic elements — roots of polynomials over K — so it adds no transcendental numbers. The complex numbers play this role for the reals: the fundamental theorem of algebra says C is algebraically closed. But C is not the algebraic closure of the rationals; that closure is the smaller field of all algebraic numbers, which leaves out transcendentals like pi.