The S5 extensions of degree 6 with minimum discriminant

The algebraic number fields of degree 6 having Galois group S-5 and minimum discriminant are determined for signatures (0, 3), (2, 2) and (6, 0). The fields F-0, F-2, F-6 are generated by roots of f(0)(t) = t(6) + 3t(4) + 2t(3) + 6t(2) + 1, f(2)(t) = t(6) - 2t(4) + 12t(3) - 16t + 8, and f(6)(t) = t(6) - 18t(4) + 9t(3) + 90t(2) - 70t - 69 respectively. Each of these fields is unique up to isomorphism. This completes the enumeration of primitive sextic fields with minimum discriminant for all possible combinations of Galois group and signature.