Inverse Galois problem for small groups

I am looking for a list of all small groups (maybe order $\leq 20$) realized as the Galois groups of a polynomial over $\mathbb{Q}$, with proof.

Any idea where I could find these? Partial answers or references would be helpful too.

Solution 1:

The best online reference I know of is A database for number fields by Jürgen Klüners and Gunter Malle. The database also provides references. (I first found the database through this MO question.)

In print, there are several excellent books related to the problem, at different degrees of accessibility. The most introductory, with many examples, is

Helmut Völklein. Groups as Galois groups. An introduction. Cambridge Studies in Advanced Mathematics, 53. Cambridge University Press, Cambridge, 1996. MR1405612 (98b:12003).

You may also want to look at

B. Heinrich Matzat. Konstruktive Galoistheorie. Lecture Notes in Mathematics, 1284. Springer-Verlag, Berlin, 1987. MR1004467 (91a:12007).

Gunter Malle, and B. Heinrich Matzat. Inverse Galois theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1999. MR1711577 (2000k:12004).

Jean-Pierre Serre. Topics in Galois theory. Second edition. With notes by Henri Darmon. Research Notes in Mathematics, 1. A K Peters, Ltd., Wellesley, MA, 2008. MR2363329 (2008i:12010). (See also MR1162313 (94d:12006) for a review of the first edition.)

Christian U. Jensen, Arne Ledet, and Noriko Yui. Generic polynomials. Constructive aspects of the inverse Galois problem. Mathematical Sciences Research Institute Publications, 45. Cambridge University Press, Cambridge, 2002. MR1969648 (2004d:12007).

All these books present specific examples together with the general theory. Generic polynomials, in particular, is concerned with constructive aspects and the question of identifying polynomials for specific groups.

As a complement, I suggest looking at these two MO questions.