Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2
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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 $74 \ \text {(ii)}$