Q:

L|K field extension and $deg(f)\nmid[L:K]$

Let L|K be finite field extension and $f \in K[X]$ is irreducible with $deg(f) > 1$. Show that, if $f\nmid[L:K]$ then f has no zeros in L.
Is it true? For ex. $f=x^3+x$ and $[Q(\sqrt 5,i):Q]$. f has zeros in $Q(\sqrt 5,i)$ and $deg (f) \nmid [Q(\sqrt 5,i):Q]$. What's wrong with my example? And how to prove it?

A:

Note that $f(x)=x^3+x$ is not a counterexample, because it's reducible.
If $\alpha\in L$ is a root of the irreducible polynomial $f(x)\in K[x]$, then $K[\alpha]$ has dimension $\deg(f)$ over $K$.
By the dimension theorem,
$$
[L:K]=[L:K[\alpha]]\,[K[\alpha]:K]
$$
so $\deg(f)$ divides $[L:K]$.

