Let $L/K$ be a field extension of degree $2^k$, $k\in\mathbb{N}$. $a\in L$ and $f\in K[X]$ a polynomial of degree $d$ such that $f(a)=0$, $d$ odd. Show that $a \in K$.
I know only the definition of a field extension and that of degree, this should be sufficient to find this. I have shown that $K[a]$ is a field and that $2^k=[L:K]=[L:K[a]][K[a]:K]$. I tried to show that $[K[a]:K]$ cannot be even, but all my arguments do not allow me to conclude.