A lot of effort is put into finding the Galois group of a polynomial. I'd like to know about what you can say when you go in the other direction.
1. Suppose I tell you that the Galois Group of a polynomial is trivial (or Z2, or something). What can I conclude from this?
2. In particular, if I know the Galois Group is trivial, am I supposed to conclude that the polynomial is / is not solvable by radicals?