# Definition:Characteristic Polynomial

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## 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 $L$.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**characteristic polynomial**