Definition:Characteristic Polynomial

Definition
Let $K$ be a field.

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

Then by 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:
 * $\operatorname{det} \left[{ X I_L - \theta_\alpha }\right]$

where:
 * $\operatorname{det}$ is the determinant of a linear operator
 * $X$ is an indeterminate
 * $I_L$ is the identity mapping on $L$.