\begin{definition}[Definition:Minimal Polynomial/Definition 3]
Let $L / K$ be a field extension.
Let $\alpha \in L$ be algebraic over $K$.
The '''minimal polynomial''' of $\alpha$ over $K$ is the unique monic polynomial $f \in K \sqbrk x$ that generates the kernel of the evaluation homomorphism $K \sqbrk x \to L$ at $\alpha$.
That is, such that for all $g \in K \left[{x}\right]$:
:$\map g \alpha = 0$ {{iff}} $f$ divides $g$.
\end{definition}