I just finished reading "A Book of Abstract Algebra" by Charles C. Pinter and, as someone who is studying this independently, I was having some understanding issues and many questions.
1) Does every polynomial of the form $a_0+a_1x+a_2x^2+...+a_nx^n$ have a Galois group that is isomorphic to $S_n$, and therefore there are no radical expressions for the roots of polynomials of degree $>4$ that have nonzero coefficients?? Does the polynomial being irreducible change the answer to the previous question?
2) The book only shows one example of a polynomial not being solvable by radicals, a quintic whose Galois group is isomorphic to $S_5$. Does every Galois group of a polynomial with complex roots have a transposition? Are the permutations of $\pm\sqrt2$ considered a transposition as well?
3) Finally, I've seen this question, the querstion, asked in some form but I've never seen it asked and answered fully. How, given a polynomial of degree n, can I determine the Galois group and therefore determine if it is solvable by radicals? From what I've read/seen, it seems to be a logical trial and error but I haven't seen any methods of doing so. Is there a method, besides trying to reason through it? If there isn't, are there any readings/exercises I can do to check if polynomials are solvable by radicals or if groups are solvable? I realize I must get more comfortable with the common group types. Any input would be great.
I apologize if this question is an amateur question, but I've only been studying this book for a couple months. I'm going to move on to Basic Algebra I to solidify my knowledge and understanding, and eventually go on to Abstract Algebra by Dummit and Foote (which I hear is the "holy grail" in this subject), but I'm open to any suggestions you might have/what you did that helped you learn Algebra.