Prolegomena

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).