Zyklische Gleichungen 6. Grades und Minimalbasis

A determination of the automorphisms of certain algebraic fields

“…(30) f()) k + akrpk q-akr[re q-k(p 2e)]k + akepr + eSr 0, where p d 2e, r 4 + 1 and a, d, e, tae on all rational values such that neither d: 4e nor 4k: + 1 is a square, is normal with the Galois group G(4, 2) provided it is irreducible; all normal octic fields with the group G (4,2) are so obtained.…”

“…In the present paper, all automorphisms are explicitly obtained by purely algebraic methods for the cyclic cubic, quartic, and sextic, the quartic with the four-group, the sextic with the symmetric group, and the octics with the Abelian groups of types (2,2,2) and (2,4). The determination of the parametric representations of the most general equations defining these fields was an integral part of the determination of the automorphisms, and while for n 3 and n 4 these results.merely confirm known facts, the purely rational method of their attainment should not be without interest.…”

Action of dihedral groups

“…On the other hand only a handful of results for the rationality of Q(G) were known before 1950's. Samson Breuer was able to show that Q(Z 3 ) and Q(Z 6 ) are rational over Q where Z n is the cyclic group of order n [Br1]; he then showed that Q(G) is rational for some transitive solvable subgroup G contained in S p , the symmetric group of degree p, if p = 5 or 7 [Br2]. Breuer's results for transitive solvable subgroups in S p was extended by Furtwängler for p = 5,7,11 [Fr]; finally Breuer himself extended these results for any prime number p ≤ 23 [Br3].…”