COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

The uses of computers in Galois theory
W. D. MAWR
ANY competent mathematician can learn FORTRAN in a few weeks and
can immediately start applying it toward solving problems which are
finite in nature. Constructing all the subgroups of a finite group, finding
the incidence numbers of a finite simplicial complex, and taking the partial
derivatives of a large symbolic expression in several variables, are examples
of such problems, which take large amounts of programming but very
little “hard” mathematical thinking. Such procedures are now widely
recognized as having great value in preliminary investigations as well as
for teaching purposes. The majority of good problems in mathematics,
however, are not finite in nature, and many mathematicians feel that the
computer is out of place in this environment. It is clear that we cannot
ask the computer to look at all cases, when the number of cases is in-finite.
Of course, in many situations, we can think of mathematical arguments
which will reduce an infinite problem to a finite problem, and this is
what is currently done in Galois theory, as detailed below. It is our hope,
however, that these mathematical arguments will eventually themselves be
generated and applied by the computer, so that the computer may be
brought directly to bear on an infinite problem.
The problem of calculating the Galois group of a polynomial over the
rationals is remarkable among mathematical algorithms for the paucity
of its input-output. A single polynomial is given as input and a single
group code, or the Cayley table of a group, is returned as output. It is
the purpose of this paper to describe the computational difficulties that
arise in computing such groups and to indicate how they may be solved.
It has been noticed several times that, although the splitting fields
whose automorphism groups are the Galois groups of polynomials over
the rationals are infinite fields, the problem of calculating these automorphism
groups is actually a finite problem. The best known statement
to this effect was made by van der Waerden in [l]. Van der Waerden’s
method of calculating a Galois group proceeds as follows: Let the polynomialfhave
degree n over the field d (in our case, d is the field of rational
numbers) and let Z be the splitting field. Consider the ring Z (ul, . . . , u,,, z)
of polynomials, with coefficients in Z, in the (n+ 1) variables ~1, . . . , u,,, z.
Form in this ring the expression 8 = arulf . . . +a,~,, where the xi are the
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