 Galois group of an irreducible even quartic

Let $g(X)=X^4+b{X}^2+c\in \mathbb{Q}[X]$ be irreducible, let $\alpha$ be a root, and put $\delta =\sqrt{c}$. Its roots areso its splitting field is $K=\mathbb{Q}(\alpha ,\delta )$. If $\delta \notin \mathbb{Q}(\alpha )$, then $[K:\mathbb{Q}]=8$ and the Galois group is $D_8$. If $\delta \in \mathbb{Q}(\alpha )$, then $[K:\mathbb{Q}]=4$ and the group is $V_4$ when $\delta \in \mathbb{Q}$, and $C_4$ otherwise.