# 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 GL_{n}(*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

## 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:^{[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 *N*_{n} from GL_{n}(*K*) to *F*^{*}. We also have a homomorphism from GL_{n}(*K*) to *F*^{*} obtained by composing the Diedonné determinant from GL_{n}(*K*) to *K*^{*}/[*K*^{*}, *K*^{*}] with the reduced norm *N*_{1} from GL_{1}(*K*) = *K*^{*} to *F*^{*} via the abelianization.

The **Tannaka–Artin problem** is whether these two maps have the same kernel SL_{n}(*K*). This is true when *F* is locally compact^{[2]} but false in general.^{[3]}

## See also

## References

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