Let $K$ be a finite extension field of a finite field $F$. Show that there is an element $a\in K$ s.t. $K = F(a)$.
My attempt:
$K$ is a finite field and $char(K) = char(F) := p$. I know that for a given prime $p$ and $n\in N$, there exists a unique finite field upto isomorphism of order $p^n$.
1) But how can I show that $K \simeq GF(p^n)$ where $p = char(K)$?
Once I have that, $K\setminus \{0\}$ is a cyclic group under multiplication and thus $\exists a\in K$, s.t. $K\setminus \{0\} = <a>\subset F(a)$. Thus $K\subset F(a)$.
2) Also, how can I show that $F(a)\subset K$ so that $F(a) = K$? $K$ is an finite field extension of $F$ but I can't understand why that would imply $F(a)\subset K$??