And $rGr^{-1}$ is a conjugate subgroup of $G$, right? So there is a 1:1 correspondence between cosets and ideals, and then some kind of onto mapping from that onto the conjugate subgroups. I think we're saying the same thing.

Thanks for your responses, Will. The only thing I still don't understand is your comment about cosets. Whichever version of the Galois ideal you look at, it is clear from the definition that the corresponding Galois group is indeed a group. And a coset is not a group. I still think the different Galois ideals must correspond to conjugate subgroups of $G \subset S_n$. In your example, as you say, $A_3 \vartriangleleft S_3$, so $A_3$ has no distinct conjugate subgroups. That makes sense. I just don't see where cosets come into play.

I guess I was confused because the trivial ideal is determined as soon as you know $f(x)$, but to find the Galois ideal you have to specify a numbering of the roots $\alpha_1, \alpha_2, ... \alpha_n$. Maybe you're telling me that there are conjugate Galois ideals corresponding to the conjugate subgroups of the Galois group $G \subset S_n$? I'm going to have to chew on this some more.

Thanks Neil-- I don't have Maple but I did check Pari, and it has a case-by-case algorithm for poly's of degree up to 11, which I found uninspiring. As mentioned, I'm really more interested in the Galois ideal than in the Galois group, and I'm not aware of anything that computes the non-trivial generators of this ideal. I am going to try to look up Eisenbud's paper on primary decomposition in Inventiones 110 if I get a chance.

OK, I see what you mean now. I will try to look up the decomposition algorithms.

OK, forgive me for being dense, but how does a primary decomposition in $\mathbb{Q}[x_1,x_2,...,x_n]$ take into account the particular polynomial $f(x)$? Can you give me an example of how this works for $f(x) = x^4 - 2$ ?