Field extensions and irreducible polynomial question

Question

I'm having some trouble solving this homework problem and understanding what the hint is trying to tell me.

Suppose $K$ is a field extension of $F$ of finite degree. Prove that if $\alpha$ is in $K$, then there is an irreducible polynomial $f(x)$ in $F[x]$ having $a$ as a root. (Hint: If $[K : F]=n$, consider $1,a,a^2,...,a^n$)

Answer 1

Here's a little hint: the family of $1,a,a^2,\dots$ generates a subring of $K$ which is also a $F$-vector space, so subspace of $K$ as $F$-vector space.

From this you could easily conclude using a little of linear algebra.