I was reading the proof of proposition 4.2 in the following book(page no 231) and stumbled upon many facts. https://books.google.ca/books?id=qWYbCAAAQBAJ&pg=PA231&lpg=PA231&dq=galois+group+of+morse+function&source=bl&ots=zQAJ5IyXqL&sig=bLcyajrJNcmSdFsB2Pvu2lPyOVw&hl=en&sa=X&ved=0ahUKEwiG9LGwg5LYAhVK0YMKHS78DDkQ6AEIQzAF#v=onepage&q=galois%20group%20of%20morse%20function&f=false
I am quoting below the parts which I have not understood and then furnishing my queries. I have some knowledge in Riemann surfaces(roughly the stuff in Otto Foster's book),basic algebraic number theory; but I do not have background in algebraic geometry. Please help me understanding the proof.
"Condition (b) says that no two critical points are in the same fibre,so an equation φ(t)=α for α∈K has at least n-1 roots in K. Therefore the map φ the simplest ramification behaviour over the affine line,the ramification group over any finite point x=α just permutes two roots of f(t)-xg(t),i.e. is generated by a transposition."
I think in the first line it means that it has at least n-1 distinct roots. I have not understood the substance of the second line:
(i) what is the ramification group at a finite point? Why in this case it is generated by a transposition?
My next doubt occurs here: "Now the affine line has a trivial tame fundamental group,i.e. there is no unramified covering of A^1 which is tamely ramified at ∞.Therefore the galois group G is generated by the ramification groups at the finite points,i.e. by transpositions."
Here my questions are:
(ii)What is this fact that A^1 has no such unramified covering ? I mean could you refer me some standard theorem.
(iii)How does it follow from the above fact that the Galois group is generated by the ramification groups at the finite points?
(iv)What is the ramification group at the infinity?
(v)Is the 'Inertia Group" at a ramified point is the same as the 'Ramification group' at that point?
I have no knowledge of the above facts and tried to read from several sources but have been confounded.Any help would be greatly appreciated.Thanks.