THE S_{5} EXTENSIONS OF DEGREE 6 WITH MINIMUM DISCRIMINANT

DAVID FORD, MICHAEL POHST, MARIO DABERKOW, AND NASSER HADDAD

Abstract. 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.