Klein-four criterion for an irreducible even quartic

For irreducible $X^4+b{X}^2+c\in \mathbb{Q}[X]$, 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 $\mathbb{Q}$.