Let $L$ be an extension of $\mathbb{Q}$, such that $[L:\mathbb{Q}]=4$ and $L$ is normal over $\mathbb{Q}$. So $L$ is splitting field of a polynomial $f(x) \in \mathbb{Q}[x]$.
My question is: Is there a way to determine the degree of $f$?
I have no another hypotesis about $f$, so for instance we can get $L = \mathbb{Q}(\sqrt{2}, \sqrt{3})$ and $f(x) = (x^2-2)(x^2-3)$ or $f(x) = (x^2-2)(x^2-3)(x-1)$. This way, it seems like $f$ could have degree whatever we want, as long as it's bigger than four. I guess we need another hypotesis, maybe $f$ must be irreducible, what do you think??
Also, I don't know if it is usefull, but since $[L:\mathbb{Q}]$ has four degree and it is normal (and of course, separable) we have $\vert Gal(L:\mathbb{Q})\vert=4$.
I would appreciate if you could help me.