Algebraic Element of Field Extension is Root of Unique Monic Polynomial of Minimal Degree
Let $L / K$ be a field extension.
Let $\alpha \in L$ be algebraic over $K$.
Let $f$ and $g$ be two such monic polynomial.
- $\map f \alpha - \map g \alpha = 0$
Aiming for a contradiction, suppose $f - g \ne 0$.
Let $a$ be the leading coefficient of $f-g$.
This is a contradiction.
Thus $f - g = 0$.