THE S-5 EXTENSIONS OF DEGREE-6 WITH MINIMUM DISCRIMINANT

The algebraic number fields of degree 6 having Galois group S-5 and mi nimum 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 respective ly. Each of these fields is unique up to isomorphism. This completes t he enumeration of primitive sextic fields with minimum discriminant fo r all possible combinations of Galois group and signature.