Dieudonné determinant

Summary

The Dieudonné determinant extends the determinant to square matrices over a division ring $D$. Its values lie in the abelianisation $D^\times/[D^\times, D^\times]$ of the multiplicative group, not in $D$ itself. It is multiplicative, and for matrices of size at least $2$ it is trivial exactly on the elementary subgroup. It gives the isomorphism $K_1(D) \cong D^\times/[D^\times, D^\times]$ for K1 of a ring.