Definition:Characteristic Polynomial of Field Extension
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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 $L$.
Also see
- Results about characteristic polynomials of field extensions can be found here.