Writing f(X) = (X− r1) · · · (X− rn), the splitting field of f(X) over K is K(r1, . . . , rn). Each σ in the Galois group of f(X) over K permutes the ri’s since σ fixes K and therefore f(r) = 0⇒ f(σ(r)) = 0. The automorphism σ is completely determined by its permutation of the ri’s since the ri’s generate the splitting field over K. A permutation of the ri’s can be viewed as a permutation of th...

In a previous work, the second-named author gave complete description of action automorphisms on ordinary irreducible characters finite symplectic groups. We generalise this in two directions. Firstly, using work first-named author, we give Galois characters. Secondly, extend both descriptions to cover case special orthogonal As consequence, obtain explicit for character fields and

Given an irreducible bivariate polynomial f (t, x) ∈ ℚ[t, x], what groups H appear as the Galois group of (t0, for infinitely many t0 ℚ? How often does a above f(t0, x), We give answer large x-degree with alternating or symmetric over ℚ(t). This is done by determining low genus subcovers coverings $$\tilde X \to \mathbb{P}_\mathbb{C}^1$$ monodromy groups.