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

Aiming for a contradiction, suppose $\alpha \in K$.

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 $\alpha \notin K$.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 38$. Simple Algebraic Extensions: Theorem $74 \ \text {(ii)}$