Let be a Galois extension, and let . The norm of is defined to be the product of all the elements of the orbit of under the group action of the Galois group on ; taken with multiplicities if is a finite extension.
In the case where is a finite extension, the norm of can be defined to be the determinant of the linear transformation given by , where is regarded as a vector space over . This definition does not require that be Galois, or even that be a field—for instance, it remains valid when is a division ring (although does have to be a field, in order for determinant to be defined). Of course, for finite Galois extensions , this definition agrees with the previous one, and moreover the formula
The norm of is always an element of , since any element of permutes the orbit of and thus fixes .
|Date of creation||2013-03-22 12:18:02|
|Last modified on||2013-03-22 12:18:02|
|Last modified by||djao (24)|