Exercise 5

Let $f(x)=x^{4}-5x^{2}+6\in\mathbb{Q}[x]$.

1. Show that one possible splitting field of $f(x)$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2})(\sqrt{3})$.

2. It is given that $p(x)=x^{4}-10x+1$ is an irreducible polynomial over $\mathbb{Q}$. Show that $p(x)$ has a zero of $\sqrt{2}+\sqrt{3}$.

3. Hence show that $\mathbb{Q}(\sqrt{2}+\sqrt{3})$ is a splitting field of $f(x)$ over $\mathbb{Q}$.