norm

Let $K/F$ be a Galois extension, and let $x\in K$. The norm ${\mathrm{N}}_F^K(x)$ 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 $\mathrm{Gal}(K/F)$ on $K$; taken with multiplicities if $K/F$ 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

$${\mathrm{N}}_F^K(x):=\prod _{\sigma \in \mathrm{Gal}(K/F)}\sigma (x)$$

holds.

The norm of $x$ is always an element of $F$, since any element of $\mathrm{Gal}(K/F)$ permutes the orbit of $x$ and thus fixes ${\mathrm{N}}_F^K(x)$.