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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.

Amoroso and Masser proved that for every real $\epsilon > 0$, there exists a constant $c(\epsilon)>0$, such that for every algebraic number $\alpha$ with $\mathbb{Q}(\alpha)/\mathbb{Q}$ being a Galois extension, the height of $\alpha$ is either 0 or at least $c(\epsilon) [\mathbb{Q}(\alpha):\mathbb{Q}]^{-\epsilon}$. In this article we establish an explicit version of this theorem.