What's the smallest Galois extension of $\mathbb{Q}(x^3)$ containing $\mathbb{Q}(x)$?
For example, let E be the smallest Galois extension of $\mathbb{Q}(x^3)$ containing $\mathbb{Q}(x)$. Then, do i have to show that $E/\mathbb{Q}(x^3)$ is a separable extension and a normal extension?