The $S_5$ extensions of degree 6 with minimum discriminant

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 $f_0(t) = t^6 + 3 t^4 + 2 t^3 + 6 t^2 + 1$, $\,f_2(t) = t^6 - 2 t^4 + 12 t^3 - 16 t + 8$, and $f_6(t) = t^6 - 18 t^4 + 9 t^3 + 90 t^2 - 70 t - 69$ 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.