Definition:Minimal Polynomial

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Definition

Let $L / K$ be a field extension.

Let $\alpha \in L$ be algebraic over $K$.


Definition 1

The minimal polynomial of $\alpha$ over $K$ is the unique monic polynomial $f \in K \left[{x}\right]$ of smallest degree such that $f \left({\alpha}\right) = 0$.


Definition 2

The minimal polynomial of $\alpha$ over $K$ is the unique irreducible, monic polynomial $f \in K \left[{x}\right]$ such that $f \left({\alpha}\right) = 0$.


Definition 3

The minimal polynomial of $\alpha$ over $K$ is the unique monic polynomial $f \in K \left[{x}\right]$ that generates the kernel of the evaluation homomorphism $K \left[{x}\right] \to L$ at $\alpha$.

That is, such that for all $g \in K \left[{x}\right]$:

$g \left({\alpha}\right) = 0$ if and only if $f$ divides $g$.


Notation

The minimal polynomial of $\alpha$ over $K$ can be denoted $\min_K \left({\alpha}\right)$ or $\mu_K \left({\alpha}\right)$.


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Note that it depends only on $\alpha$ and $K$.


Also see