On the height of some generators of galois extensions with big galois group

We study the height of generators of Galois extensions of the rationals having the alternating group ${\mathfrak{A}}_n$ as Galois group. We prove that if such generators are obtained from certain, albeit classical, constructions, their height tends to infinity as $n$ increases. This provides an analogue of a result by Amoroso, originally established for the symmetric group.