Definition:Field Norm
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Definition
Let $K$ be a field and $L/K$ a finite 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 field norm $N_{L/K}(\alpha)$ of $\alpha$ is the determinant of this map.
That the field norm is in fact a norm is proved in Field Norm is a Norm.
I think the field $K$ needs to be assigned a norm explicitly to be able to verify this is a norm
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