Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2

Theorem
Let $K / F$ be a finite field extension of degree $2^m$.

Let $\alpha \in K$ be algebraic over $F$ with degree $3$.

Then $\alpha \notin K$.

Proof
$\alpha \in K$.

From Degree of Element of Finite Field Extension divides Degree of Extension:
 * $\map \deg \alpha \divides \map \deg {K / F}$

But:
 * $3 \nmid 2^m$

From this contradiction, it follows that $\alpha \notin K$.