I feel like this fact should be easy but I'm struggling to see it. If I have a polynomial $f \in K[x]$ which is irreducible and has roots $\alpha$, $\beta$ in some finite normal (over $K$) extension field $L$, is it true that $L$ is then the splitting field of $f$ over $K(\alpha)$?
I know that in general a finite extension is normal if and only if it is the splitting field of SOME polynomial. I also know that if $L$ is normal over $K$ then it is normal over $K(\alpha)$, hence $f \in K(\alpha)[x]$ must split in $L$, but how do we know there is no intermediate field $A$ between $K(\alpha)$ and $L$ where $f$ also splits?
Thanks
Best Answer
In fact you don't know that there are no intermediate fields between $K(\alpha)$ and $L$. Take as an example $K = \mathbb{Q}$, $L =$ splitting field of $x^4+1$. Then $f=x^2+1 \in \mathbb{Q}[x]$ is irreducible, it factors inside $L$, but $L$ is NOT the splitting field of $f$ over $\mathbb{Q}$.