Dear Gap Forum,

Igor Schein asked:

using Galois() function in Gap3, I can find a Galois group of a
irreducible polynomial. Is there a way to accomplish an opposite,
i.e. given a finite solvable group, find a polynomial which would have
this group as its Galois group.

This depends a little bit on how you want to accomplish it and how much work
you are willing to invest.

As far as I know the original proof by Safarejevic that every solvable group
occurs as a Galois group over the rationals does not give an explict
construction, nor have I ever encountered a general construction for
solvable groups that is explicit.

The best I've seen in this direction is a proof for the special case of
nilpotent groups in the book by Serre on the inverse problem of Galois
theory.

There are explicit lists of examples for (solvable) groups of small degrees
(computed, among others, by Eichenlaub, Kl"uners, Malle, Matzat, Smith,...)
but creating these lists involves hard polynomial calculations and is rather
an art than an general purpose algorithm.
(If you are interested I can give you further pointers to the literature.)

This also implies that there is no CA system, that provides a function to
plug in a group and obtain a polynomial.

Best regards,

Alexander Hulpke