Square-free discriminants and affect-free equations (Q810578)

Let K be an algebraic number field of degree n and let \(\bar K\) denote the Galois closure of \(K/{\mathbb{Q}}\). Suppose that the discriminant d of K is square-free. It is proved that the Galois group of \(\bar K/{\mathbb{Q}}\) is the symmetric group \(S_ n\), and that every prime ideal is unramified in \(\bar K/{\mathbb{Q}}(\sqrt{d})\). As an application, it is shown that certain trinomial equations are affect-free.