This may not initially be a well posed question, but I'm looking for a good reference on Galois theory that covers it from the viewpoint of the symmetry in roots of an irreducible polynomial and not a field extension. It's well known that a generic polynomial has Galois group $S_n$, and this can be interpreted as saying the only relations satisfied by the roots are expressed by symmetric polynomials and the coefficients of the original polynomial (eg, the constant term is the product of all roots). Then to get a smaller group, one has extra implicit relations among the roots. (I'm having trouble finding an example to even look at right now, but in the case of deg 3, you would have a polynomial whose Galois group is cyclic so you get three relations as above, but there must be one other relation which is not symmetric under an element of order two in $S_3$.) Can we give some criteria to determine when you have extra relations, or, if you know the group is not all of $S_n$, can we say what these relations are? (I like to think of this as cutting down the size of the Galois group by adding equations - similar to intersecting hypersurfaces, say.)
Another way to think of this is that, for fixed degree $n$, we get a stratification of $\mathbb{Q}^{n-1}$, (or your favorite field) by associating to $(a_1, \ldots, a_{n-1})$ the Galois group of $x^n + a_1 x^{n-2} + \ldots + a_{n-2} x + a_{n-1}$. Would this stratification be 'nice'?