= Galois group of an irreducible even quartic
{c}
{title2=$X^4+bX^2+c$}
Let $g(X)=X^4+bX^2+c\in\mathbb Q[X]$ be irreducible, let $\alpha$ be a root, and put $\delta=\sqrt c$. Its roots are
$$
\alpha,-\alpha,\frac\delta\alpha,-\frac\delta\alpha,
$$
so 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.
Back to article page
