i have found several solutions over the net, the one i want to understand uses Hilbert's irreducibility theorem whose proof i understood more or less , but my problem is with the solution now. this proof is going as to prove that $f (x, y) = y^n - xy - x$ in $Q[x, y]$ has Galois group $G = S_n$ over $Q(x)$, and then we can use Hilbert's irreducibility theorem.
1st step : for any $r \in Q$, the Galois group of $f (r, y) \in Q[ y]$ over $Q$ injects canonically in the Galois group $G$ of $f(x,y)$ over $Q(x)$. what is the mapping?
2) $f(y)$ is irreducible over $Q(x)$ by Eisenstein’s criterion?? how?? Eisenstein criterion determines irreducibility of polynomials over UFD where there is a prime element s.t is divides last n-1 coefficients but not first and its square also doesn't divide last one. now $f(y)\in Q[x][y]$ so what is the prime element in $Q[x]$ which satisfies Eisenstein criterion for $f(y)$??
3) show that $G$ acts doubly transitively on the roots. ( any hint for this will work)
please help me, give some hints, i really need to do this, understand this.
first please help me understand these two things... rest is one more important step which i will post later after understanding this. please help...