Example 4

Let $D'$ be a field, and let $u$ be a homomorphism from D to $D'$. When passing to the quotients, $u$ defines a group homomorphism $u_{ab}$ from $D^*_{ab}$ to $D^{'*}_{ab}$. Let $u_n$ be the homomorphism from $\mathbf{G}\mathbf{L}_n(D)$ to $\mathbf{G}\mathbf{L}_n(D')$ that transforms a matrix A $= (a_{ij})$ into the matrix $(u(a_{ij}))$. The formula

(23) det($u_n($A)) $=u_{ab}$(det(A))

for $A\in \mathbf{G}\mathbf{L}_n(D)$ follows immediately from Example 3.