Degree of an extension field

Question

Suppose that I have a finite field $GF(2)$. Then, suppose to extend it to $GF(2^{13})$. Finally, consider a irreducible polynomial of degree $256$ over $GF(2^{13})$. What is the degree of such extension?

Also, how are structured the elements of this extension?

Thank you

Answer 1

Hint: If $F$ is a field and $f$ is an irreducible polynomial of degree $n$ over $F$, then $E=F[X]/(p)$ is an extension of $F$ of degree $n$.

If $F$ is an extension of $K$ of degree $m$, then $E$ is an extension of $K$ of degree $mn$.