For any integer $m\neq 0$, we prove that $f(x)=x^{9}+9mx^{6}+192m^{3}$ is irreducible over $\mathbb{Q}$ and that the Galois group of $f(x)$ over $\mathbb{Q}$ is the dihedral group of order 18. Moreover, we show that for infinitely many values of $m$, the splitting fields for $f(x)$ are distinct.

Let K be a field of characteristic 0, which is algebraically closed to radicals. Let F = K((X)) be the valued field of Laurent power series and let G = Aut(F/K). We prove that if L is a subfield of F, K ≠ L, such that L/K is a sub-extension of F/K and F/L is a Galois algebraic extension (L/K is Galois coalgebraic in F/K), then L is closed in F, F/L is a finite extension and Gal(F/L) is a finite cyclic group of G. We also prove that there is a one-to-one and onto correspondence between the set of all finite subgroups of G and the set of all Galois coalgebraic sub-extensions of F/K. Some other auxiliary results which are useful by their own are given.

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.