Consider the polynomial $f(x)=x^4+6x^3+32x^2+17x-15$ and let $\alpha\in\mathbb{C}$ be a root of $f$. How can I show that $\mathbb{Q}(\alpha)$ has no subfield of degree 2 over $\mathbb{Q}$? I have an hint: consider the Galois group of $f$ over $\mathbb{F}_2$ and $\mathbb{F}_3$.
The Galois group over $\mathbb{F}_2$ is $\mathbb{Z}/4\mathbb{Z}$ and over $\mathbb{F}_3$ is $\mathbb{Z}/3\mathbb{Z}$, so $G:=Gal_\mathbb{Q}(f)$ has a 4-cycle and a 3-cycle (and of course $G