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 group homomorphism from the group GL_n 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 the residue class, in K^×/, of

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:

The determinant is invariant under elementary row operations
The determinant of the identity matrix is 1
If a row is left multiplied by a in R^× then the determinant is left multiplied by a
The determinant is multiplicative: det = detdet
If two rows are exchanged, the determinant is multiplied by −1
If R is commutative, then the determinant is invariant under transposition

Tannaka–Artin problem

Assume that K is finite over its center F. The reduced norm gives a homomorphism N_n from GL_n to F^×. We also have a homomorphism from GL_n to F^× obtained by composing the Dieudonné determinant from GL_n 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 SL_n. This is true when F is locally compact but false in general.