Let $K/F$ be a Galois extension, and let $x \in K$ . The 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.

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 $$ \operatorname{N}_F^K(x) := \prod_{\sigma \in \operatorname{Gal}(K/F)} \sigma(x) $$ holds.

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 $\operatorname{N}_F^K(x)$ .