Infinite families of \(A_4\)-sextic polynomials (Q2925371)

The authors prove that if \(g(X) \in \mathbb{Z}[X]\) is a monic cubic polynomial such that \(f(X):=g(X^2)\) is irreducible over \(\mathbb{Q}\) and the discriminants of both \(f\) and \(g\) are squares in \(\mathbb{Z}\), then the Galois group of \(f\) is isomorphic to the alternating group \(A_4\). They use their test to obtain three infinite families of sextic polynomials irreducible over \(\mathbb{Q}\) with Galois group isomorphic to \(A_4\).