The extension field K of a field F is called a splitting field for the polynomial f(x) element F[x] if f(x) factors completely into linear factors in K[x] and f(x) does not factor completely into linear factors over any proper subfield of K containing F. For example, the extension field Q(sqrt(3)i) is the splitting field for x^2 + 3 since it is the smallest field containing its roots, sqrt(3)i and -sqrt(3)i. Note that it is also the splitting field for x^3 + 1.