how to show Sn is realizable over Q? i.e as a galois group of some polynomial over Q?

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 = Sn over Q(x), and then we can use Hilbert's irreducibility theorem.

1st step : for any r ∈ Q, the Galois group of f (r, y) ∈ Q[ y] over Q injects canonically in

the Galois group 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 blah blah...

f(y)∈ Q[x][y] so what is the prime element in Q[x] which satisfies eisenstein criterion for f(y)??

first please help me understand these two things... rest is one more important step which i will post later after understanding this. please help...