Klein-four criterion for an irreducible even quartic
= Klein-four criterion for an irreducible even quartic
{c}
For irreducible $X^4+bX^2+c\in\mathbb Q[X]$, the Galois group is $C_2\times C_2$ exactly when $c$ is a square in $\mathbb Q$. In the transitive Klein-four action on the roots $\alpha,-\alpha,\beta,-\beta$, all three nonidentity double transpositions fix $\alpha\beta=\sqrt c$, forcing it into the fixed field $\mathbb Q$.