Definition:Characteristic Polynomial

Definition
Let $K$ be a field and $L/K$ 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 map:


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

The characteristic polynomial of $\alpha$ with respect to the extension $E/F$ is $\operatorname{det}\left[ XI - \theta_\alpha \right]$.

Here $\operatorname{det}$ is the determinant of a linear map, $X$ is an indeterminate and $I$ is the identity on $L$.