\begin{definition}[Definition:Characteristic Polynomial]
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 $L$.
\end{definition}