Category:Examples of Minimal Polynomials

From ProofWiki
Jump to navigation Jump to search

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 $g \in K \left[{x}\right]$:

$\map g \alpha = 0$ if and only if $f$ divides $g$.

Pages in category "Examples of Minimal Polynomials"

The following 2 pages are in this category, out of 2 total.