The Determination of Galois Groups

TL;DR: In this article, a technique for computing the Galois groups of polynomials with integer coefficients is described, which can be used to determine the order of the polynomial roots.

Abstract: A technique is described for the nontentative computer determination of the Galois groups of irreducible polynomials with integer coefficients. The technique for a given polynomial involves finding high-precision approximations to the roots of the poly- nomial, and fixing an ordering for these roots. The roots are then used to create resolvent polynomials of relatively small degree, the linear factors of which determine new orderings for the roots. Sequences of these resolvents isolate the Galois group of the polynomial. Machine implementation of the technique requires the use of multiple-precision integer and multiple-precision real and complex floating-point arithmetic. Using this technique, the writer has developed programs for the determination of the Galois groups of polynomials of degree N _ 7. Two exemplary calculations are given. Introduction. The existence of an algorithm for the determination of Galois groups is nothing new; indeed, the original definition of the Galois group contained, at least implicitly, a technique for its determination, and this technique has been described explicitly by many authors (cf. van der Waerden (8, p. 189)). These sources show that the problem of finding the Galois group of a polynomial p(x) of degree n over a given field K can be reduced to the problem of factoring over K a polynomial of degree n! whose coefficients are symmetric functions of the roots of p(x). In principle, therefore, whenever we have a factoring algorithm over K, we also have a Galois group algorithm. In particular, since Kronecker has described a factoring algorithm for polynomials with rational coefficients, the problem of determining the Galois groups of such polynomials is solved in principle. It is obvious, however, that a procedure which requires the factorization of a polynomial of degree n! is not suited to the uses of mortal men. In the next sections we describe a practical and relatively simple procedure which has been used to develop programs for polynomials of degrees 3 through 7. Restrictions. The algorithm to be described will apply only to irreducible monic polynomials with integer coefficients. Since any polynomial with rational coefficients can easily be transformed into a monic polynomial with integer coefficients equivalent with respect to its Galois group, these latter two adjectives create no genuine restric- tion. The irreducibility restriction is genuine, however. For suppose p(x) = p,(x)p2(x), and suppose K1 and K2 are the splitting fields of P, and p2, respectively. If K1 n K2 = the rationals, then the Galois group of p(x) is the direct sum of the Galois groups of p,(x) and p2(x), and there is no difficulty. If, on the other hand, K1 n K2 is larger than the rationals, then the group of p(x) is not easily determined from those of p,(x) and p2(x) without explicit knowledge of the relations which exist between the