I'm trying to show that all possible splitting fields occur for a class of cubic polynomials, so have started by looking at the discriminants.

Clearly, given that they are products of squares, all discriminants are congruent to 0 or 1 mod 4. And it is easy to show that all numbers of this form appear as discriminants of quadratics. But this does not seem to be the case for discriminants of cubics.

One would think that this would have been determined, but I can't seem to find any mention of it in any literature. After a lengthy search, I can only produce about a third of all numbers 0 or 1 mod 4 lying between -100 and 100.

Does anyone know of any other restrictions? Am I missing something obvious?!