Splitting Field

A splitting field of a polynomial p(X) over a field K is a field extension L of K over which p factors into linear factors

and such that the coefficients a_i generate L over K. The extension L is then an extension of minimal degree over K in which p splits. 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).