Galois theory is a standard topic in every algebra course. Computational and constructive methods in Galois theory have not yet attained this status.
Algorithms to compute Galois groups go back as far as the nineteenth century and are described in the classical monograph of TschebotarΓΆw and Schwerdtfeger (1950). A new chapter of computational Galois theory began with the use of computers. This is marked by the article of Stauduhar (1973) giving an explicit algorithm for computing the Galois group of polynomials up to degree 7 (and 8 in his thesis), based on subgroup descent and numerical approximation of the roots. Recent implementations are available with PARI up to degree 11 and KANT up to degree 15. The method is now also explained in the monograph of Cohen (1993) and in the recent algebra textbook of Stroth (1998).
The resolvent method proposed by Soicher and McKay (1985) and further developed by Mattman and McKay (1997) avoids the precision problem caused by the use of numerical approximations of the roots. Here instead resolvent polynomials have to be factorized whose coefficients can be computed directly from the coefficients of the polynomial (Casperson and McKay, 1994). The resulting algorithm up to degree 8 is now implemented and distributed with MAPLE V. A theoretically interesting variant based on symbolic evaluation of invariants is proposed by Colin (1995).
A further method to overcome the precision problem of the Stauduhar method is padic approximation. This has been used on examples by Darmon and Ford (1989) and developed to an algorithm by Yokoyama (1997). A refined algorithm and an effective implementation up to degree 15 have been worked out by Geissler and KlΓΌners. These appear in this issue.
Apart from the algebraic methods mentioned above, in the case of geometric (regular) Galois extensions of function fields of characteristic zero, the monodromy method based on topological considerations can be of use. This is not considered here since it fits better into the context of differential Galois theory. A first satisfying implementation goes back to Deconinck and van Hoeij and is available in MAPLE V.6 as part of the algcurves package. For a more detailed discussion of recent developments in computational Galois theory see the survey article by Hulpke (1999).
Another question of interest is the inverse Galois problem. It is still not known whether every finite group is a Galois group over a given field such as Q. In particular, there does not yet exist a general method which allows one to construct, for a given group G, a Galois extension over Q, say, with group G. However, there are several strategies to prove existence and compute generating polynomials, see the recent monograph by Malle and Matzat (1999) for the state of the art.
For the realization of simple and quasi-simple groups as Galois groups the rigidity