I'm answering with a stub that I hope others will fill out. Please feel free to edit this.

The rich structure on "the solution set of a polynomial" can mean the nice properties of the function which assigns to a polynomial its multiset of roots rather than a value of that function. It can mean that we have a permutation representation of the absolute Galois group on the roots.

The rich structure on the absolute Galois group may describe the actions on many objects, the lattice of (normal) subgroups, and that it carries a topology as a profinite group. Perhaps to say an object has a rich structure in this sense, there should be natural morphisms to many interesting objects, and we can recognize it as an image of a forgetful functor.