$F$ - algebraically closed field. $k$ is a subfield of $F$. The dimension of $F$ over $k$ is a finite number $n$.
GENERAL QUESTION: What power can have then irreducible polynomials over $k$?
My attempts to solve this task:
1) Let's consider the irreducible polynomial $f$ (which has a solution $l$ in $F$).
2) Look at $k[l]$ - subring (and it is the same field), that generate all elements of $k$ and $l$ - one solution of $f$ in $F$.
3) I'll take a polynom $g$ and divide with remainder it into my polinom $f$.
QUESTION: if i take information about this subring $k[l]$ I'll know about powers of my polynoms in $k$.
Can anybody help me with an idea, please? Thank you in advance!