State (without proof) a result concerning uniqueness of splitting fields of a polynomial.

Given a polynomial $f \in \mathbb{Q}[X]$ with distinct roots, what is meant by its Galois group $\operatorname{Gal}_{\mathbb{Q}}(f)$ ? Show that $f$ is irreducible over $\mathbb{Q}$ if and only if $\mathrm{Gal}_{\mathbb{Q}}(f)$ acts transitively on the roots of $f$.

Now consider an irreducible quartic of the form $g(X)=X^{4}+b X^{2}+c \in \mathbb{Q}[X]$. If $\alpha \in \mathbb{C}$ denotes a root of $g$, show that the splitting field $K \subset \mathbb{C}$ is $\mathbb{Q}(\alpha, \sqrt{c})$. Give an explicit description of $\operatorname{Gal}(K / \mathbb{Q})$ in the cases:

(i) $\sqrt{c} \in \mathbb{Q}(\alpha)$, and

(ii) $\sqrt{c} \notin \mathbb{Q}(\alpha)$.

If $c$ is a square in $\mathbb{Q}$, deduce that $\operatorname{Gal}_{\mathbb{Q}}(g) \cong C_{2} \times C_{2}$. Conversely, if Gal $_{\mathbb{Q}}(g) \cong$ $C_{2} \times C_{2}$, show that $\sqrt{c}$ is invariant under at least two elements of order two in the Galois group, and deduce that $c$ is a square in $\mathbb{Q}$.