Galois Groups through Invariant Relations

Let f ∈ Q[x] be an irreducible polynomial of degree n. Then the splitting field L ≥ Q of f is a normal extension. We want to determine the Galois group G = Gal(f) = Gal(L/Q) of f which is the group of all field automorphisms of this extension. This task is basic in computational number theory [Coh93] as the Galois group determines a lot of properties of the field extension defined by f . Because the index [L : Q] might be large, however, we do not intend to construct L and thus cannot give explicitly the automorphism action of G on L. Instead we consider the action of G on the roots {α1, . . . , αn} of f . As f is defined over the rationals the set of these root must remain invariant under G. This permutation action is faithful, because L can be obtained by adjoining all the αi to Q. This action has to be transitive because f is irreducible. In other words: For a fixed arrangement of the roots, G can be considered as a transitive subgroup of Sn. We will utilize this embedding without explicitly mentioning it. While the problem itself is initially number-theoretic, the approaches to solve it are mainly based on commutative algebra and permutation group theory. This paper presents a new approach, approximating the Galois group by its closures (subgroups of Sn that stabilizer orbits of G). This in turn gives rise to questions about permutation groups. In the course of the paper we shall need a few facts from number theory about p-adic extensions and the relation between extensions of Q and extensions of Fp. These will be provided in an appendix.