Hermez and Salinier, Rational trinomials with the [alternating group][1] as Galois group, Journal of Number Theory, Volume 90, Issue 1, September 2001, Pages 113-129 has the abstract, 

For any integer $n\ge7$, we show how to explicitly build an infinite number of rational trinomals of degree $n$ whose Galois group over $\bf Q$ is isomorphic to $A_n$.


  [1]: https://www.sciencedirect.com/science/article/pii/S0022314X01926538