# 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 $L$.