Double covers of S5 and Frobenius groups as Galois groups over number fields

Galois Groups over Nonrigid Fields

Let F be a field with charF 6= 2. We show that there are two groups of order 32, respectively 64, such that a field F with char F 6= 2 is nonrigid if and only if at least one of the two groups is realizable as a Galois group over F . The realizability of those groups turns out to be equivalent to the realizability of certain quotients (of order 16, respectively 32). Using known results on conne...

Minimal Discriminants for Fields with Small Frobenius Groups as Galois Groups

Computation of Galois groups over function fields

Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1, t2, . . . , tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.

On projective linear groups over finite fields as Galois groups over the rational numbers

Ideas and techniques from Khare’s and Wintenberger’s article on the proof of Serre’s conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL2(Flr ) (for r running) occur as Galois groups over the rationals such that the corresponding number fields are unramified outside a set consisting of l, the infinite place and only one other prime.

Actions of Galois Groups on Invariants of Number Fields

In this paper we investigate the connection between relations among various invariants of number fields L coresponding to subgroups H acting on L and of linear relations among norm idempotents.