# Definition:Characteristic Polynomial

 It has been suggested that this page or section be merged into Definition:Characteristic Polynomial of Element of Algebra. (Discuss)

## Definition

Let $K$ be a field.

Let $L / K$ be a finite field extension of $K$.

Then by Vector Space on Field Extension is Vector Space, $L$ is naturally a vector space over $K$.

Let $\alpha \in L$, and $\theta_\alpha$ be the linear operator:

$\theta_\alpha: L \to L : \beta \mapsto \alpha \beta$

The characteristic polynomial of $\alpha$ with respect to the extension $L / K$ is:

$\det \sqbrk {X I_L - \theta_\alpha}$

where:

$\det$ denotes the determinant of a linear operator
$X$ is an indeterminate
$I_L$ is the identity mapping on $L$.