Splitting field

In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X - a_i, and such that the a_i generate L over K. It can be shown that such splitting fields exist, and are unique up to isomorphism; the amount of freedom in that isomorphism is known to be the Galois group of P (if we assume it is separable, anyway).

Given an algebraically closed field A containing K, there is a unique splitting field L of P between K and A, generated by the roots of P. Therefore, for example, for K given as a subfield of the complex numbers, the existence is automatic. On the other hand the existence of algebraic closures in general is usually proved by 'passing to the limit' from the splitting field result; which is therefore proved directly to avoid a vicious circle.