again a splitting field question

[fredbel6]

hi, i am trying to gain as much experience possible in finding degrees and (if possible) automorphism groups of splitting field extensions of separable polynomials , especially over Q

i now know these theorems :

an irreducible polynomial over a field (with characteristic not two) and degree three, has splitting field with degree three if and only iff the discriminant is square in the groundfield

an irreducible polynomial x^4+a*x*x+b over a field (not char two either) has
if b is square : degree four , and the automorphism group is the dihedral group
if b is not square but (a*a-4*b)*b is square: degree four and cyclic
if b and (a*a-4*b)*b is not square : degree eight, dihedral group on four elements

but i'd like to have some examples of four degree polynomials (irreducible) with degree twelve and degree 24

can every finite group occur as an galois group?

are there more general methods i need to know?

[Peter Scholze]

[3X.lich]

[Peter Scholze]

ah sorry, i mixed something up, sorry again(you have to replace galois group by ideal class group in the above post). at least i know that for every finite abelian group G there exist number fields L, K s.t. L is unramified over K and Gal(L/K)=G.

see http://mathworld.wolfram.com/GaloisGroup.html: it seems like it's true that it's an open problem whether every finite group occurs as the galois group of some number field.