[[Field extension]]
# Field norm

Let $L:K$ be a [[field extension]] of finite degree $n$,
and identify $L$ with $K^n$
For any $\gamma \in K$ let $\Lambda(\gamma)$ denote the left-multiplication operator.
The **field norm** $N_{L:K}(\gamma)$ is defined as the [[Determinant]] of $\Lambda(\gamma)$ #m/def/ring
$$
\begin{align*}
N_{L:K} : L &\to K \\
\gamma &\mapsto \det \Lambda(\gamma)
\end{align*}
$$

## Special cases

- [[Norm of a number field]]

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#state/develop | #lang/en | #SemBr