Category:Examples of Minimal Polynomials

This category contains examples of Minimal Polynomial.

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 $f \in K \sqbrk x$ of smallest degree such that $\map f \alpha = 0$.

Definition 2

The minimal polynomial of $\alpha$ over $K$ is the unique irreducible, monic polynomial $f \in K \sqbrk x$ such that $\map f \alpha = 0$.

Definition 3

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 $g$.