# 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 Template:Harvs.

If K is a division ring, then the Dieudonné determinant is a homomorphism of groups 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

${\displaystyle \det \left({\begin{array}{*{20}c}a&b\\c&d\end{array}}\right)=\left\lbrace {\begin{array}{*{20}c}-cb&{\text{if }}a=0\\ad-aca^{-1}b&{\text{if }}a\neq 0\end{array}}\right..}$

## Properties

Let R be a local ring. There is a determinant map from the matrix ring GL(R) to the abelianised unit group Rab with the following properties:[1]

• The determinant is invariant under elementary row operations
• The determinant of the identity 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
• The determinant is invariant under transposition

## Tannaka–Artin problem

Assume that K is finite over its centre 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 Diedonné 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[2] but false in general.[3]

## References

1. Rosenberg (1994) p.64
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