Definition:Minimal Polynomial/Definition 3
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Definition
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 $g \in K \sqbrk x$:
- $\map g \alpha = 0$ if and only if $f$ divides $g$.