Let K/F be a Galois extensionMathworldPlanetmath, and let xK. The norm NFK(x) of x is defined to be the product of all the elements of the orbit of x under the group actionMathworldPlanetmath of the Galois groupMathworldPlanetmath Gal(K/F) on K; taken with multiplicities if K/F is a finite extensionMathworldPlanetmath.

In the case where K/F is a finite extension, the norm of x can be defined to be the determinantDlmfMathworldPlanetmath of the linear transformation [x]:KK given by [x](k):=xk, where K is regarded as a vector spaceMathworldPlanetmath 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



The norm of x is always an element of F, since any element of Gal(K/F) permutes the orbit of x and thus fixes NFK(x).

Title norm
Canonical name Norm
Date of creation 2013-03-22 12:18:02
Last modified on 2013-03-22 12:18:02
Owner djao (24)
Last modified by djao (24)
Numerical id 5
Author djao (24)
Entry type Definition
Classification msc 12F05