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Version 1 |
| Let $K/F$ be a Galois extension, and let $x \in K$. The {\em norm} $\operatorname{N}_F^K(x)$ of $x$ is defined to be the product of all the elements of the orbit of $x$ under the group action of the Galois group $\operatorname{Gal}(K/F)$ on $K$; taken with multiplicities if $K/F$ is a finite extension. |
Let $K/F$ be a Galois extension, and let $x \in K$. The {\em norm} $\operatorname{N}_F^K(x)$ of $x$ is defined to be the product of all the elements of the orbit of $x$ under the group action of the Galois group $\operatorname{Gal}(K/F)$ on $K$; taken with multiplicities if $K/F$ is a finite extension. |
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In the case where $K/F$ is a finite extension, |
| In the case where $K/F$ is a finite extension, the norm of $x$ can be defined to be the determinant of the linear transformation $[x]: K \to K$ given by $[x](k) := xk$, where $K$ is regarded as a vector space over $F$. This definition does not require that $K/F$ be Galois, or even that $K$ be a field---for instance, it remains valid when $K$ is a division ring (although $F$ does have to be a field, in order for determinant to be defined). Of course, for finite Galois extensions $K/F$, this definition agrees with the previous one, and moreover the formula |
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| $$ |
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| \operatorname{N}_F^K(x) := \prod_{\sigma \in \operatorname{Gal}(K/F)} \sigma(x) |
\operatorname{N}_F^K(x) := \prod_{\sigma \in \operatorname{Gal}(K/F)} \sigma(x) |
| $$ |
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| holds. |
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| The norm of $x$ is always an element of $F$, since any element of $\operatorname{Gal}(K/F)$ permutes the orbit of $x$ and thus fixes $\operatorname{N}_F^K(x)$. |
The norm of $x$ is always an element of $F$, since any element of $\operatorname{Gal}(K/F)$ permutes the orbit of $x$ and thus fixes $\operatorname{N}_F^K(x)$. |
