Alternating groups as galois groups

Question

Is there an elementary proof that the alternating group $A_n$, for any $n$, is the Galois group of an extension of the rationals? In fact, I am looking for a proof which does not use Hilbert's irreducibility theorem. This can be done for $S_n$ by establishing the existence of an irreducible polynomial in $\mathbb Z[X]$, of degree $n$, whose Galois group over the rationals contains a transposition and an $(n-1)$-cycle. I have been wondering whether an analogous proof could be found for $A_n$.