Dieudonné determinant


In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by.
If K is a division ring, then the Dieudonné determinant is a homomorphism of groups from the group GLn of invertible n by n matrices over K onto the abelianization K×/ of the multiplicative group K× of K.
For example, the Dieudonné determinant for a 2-by-2 matrix is

Properties

Let R be a local ring. There is a determinant map from the matrix ring GL to the abelianised unit group R×ab with the following properties:
Assume that K is finite over its centre F. The reduced norm gives a homomorphism Nn from GLn to F×. We also have a homomorphism from GLn to F× obtained by composing the Dieudonné determinant from GLn to K×/ with the reduced norm N1 from GL1 = K× to F× via the abelianization.
The Tannaka–Artin problem is whether these two maps have the same kernel SLn. This is true when F is locally compact but false in general.