Reduced norm

Summary

The reduced norm is a multiplicative map $\mathrm{Nrd} \colon A \to F$ on a central simple algebra $A$ over a field $F$. It is the determinant of the matrix of an element after the algebra is split into a matrix algebra over a larger field. An element is invertible exactly when its reduced norm is nonzero. The kernel of the reduced norm on $K_1(A)$ is the group SK1 of the algebra.