Let $f(x)=x^2+ax+b$ and let $K$ be the splitting field of $f(x^2)$ over $\mathbb{Q}$. Show that $Gal(K/\mathbb{Q})\leq D_8$ where $D_8$ is a dihedral group with 8 elements.
I feel like there should be more information. What I have tried is to classify all possible situation, for example, when $f(x^2)$ split completely, when $f(x^2)$ is irreducible, when $f(x^2)$ is factored into two quadratic polynomials and when $f(x^2)$ factored into one linear and one cubic. But I always ended up to use discriminant or resolvant cubic which seems too arbitrary considering arbitrary $a$ and $b$. Is this problem possible to be proved? Thanks in advance!