This is some questions I still have after I visit the Galois group theory text book several times. I think I need to revisit it again to clarify these questions. but if you do have some comments on how should I clarify these questions, that will be appreciated
why we study the F-automorphism in the first place ? is it because the symmetry among the roots that we are going to visit ? I am still not quite clear about this concept
why is the extension field and Galois group connected ? is it because the symmetry among the roots ? some roots exhibit certain symmetry in certain extension field that can be described by Galois group.
in order for an polynomial equation to be radically solvable, the Galois group has this kind structure: $G_0 <(included in ) G_1 < G_2 ... < G_n $. why $G_k$ need to be normal and abelian ? I can somewhat understand the normal constraint: the symmetry in smaller Galois group is only fixed within corresponding extension field, when view from a broader extension field this sub-symmetry no longer holds for the broader extension filed. it actually fixes roots that in this broader extension filed.
yes, I might have mis-understandings, please do help point those out and give some suggestons what I should concentrate on when I revisit the text-book