Minimal discriminants for fields with small Frobenius groups as Galois groups. (Q1874326)

The authors compute the minimum discriminants of certain number fields of small degree which have a solvable Galois group. In detail, they provide minimum discriminants and - in almost all cases - generating polynomials for (i) octic fields which contain a quadratic subfield but no quartic subfield (the relative Galois group is isomorphic to \(A_4\) or \(S_4\)); (ii) fields with Galois group isomorphic to \(C_p \rtimes C_l\) for \(p \in \{7,11,13\}\) and all \(l>1\) dividing \(p-1\); (iii) two primitive octic fields with a solvable Galois group. The tools used are essentially computational methods from class field theory.