In teaching an abstract algebra course, including elementary Galois
theory, I needed to come up with examples of field extensions of the
rational numbers whose Galois groups were particular groups, groups
whose structure we had already studied. But it was not easy for me to
find Galois extensions which had given non-Abelian Galois groups. After
some inquiries, I was told that this problem of determining whether
there are finite extensions of the field of rational numbers with given
Galois groups was first raised by Emmy Noether and that the problem is
still unsolved in general. Can anybody give me pointers to literature
on this subject. In particular, I would like to find a list of
extension fields of the rationals corresponding to some "small" groups,
say, all groups of order up to 16. Thanks for any help you can give me.
Victor Katz