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 GLn(K) of invertible n-by-n matrices over K onto the abelianization K×/[K×, 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×/[K×, K×], of
Properties Let R be a local ring. There is a determinant map from the matrix ring GL(R) to the abelianised unit group R×ab with the following properties:Rosenberg (1994) p.64 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(AB) = det(A)det(B) 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 Nn from GLn(K) to F×. We also have a homomorphism from GLn(K) to F× obtained by composing the Dieudonné determinant from GLn(K) to K×/[K×, K×] with the reduced norm N1 from GL1(K) = K× to F× via the abelianization.
The Tannaka–Artin problem is whether these two maps have the same kernel SLn(K). This is true when F is locally compact but false in general.
See also Moore determinant over a division algebra
References
. Errata