definition of reduced trace and reduced norm

From a general theory of central simple algebra, we have a notion of reduced traces and reduced norms. (See for example [1] for a theory of reduced norms and reduced traces.)

It may be defined as follows. $D_n(k)$ is a central simple algebra over $K_n(k)$. Let $\Omega$ be a splitting field $\Omega$ of $D_n(k)$. That means, we have an isomorphism of $k$-algebras

$$\iota: D_n\otimes_{K_n(k)} \Omega \cong M_{p^{n}}(\Omega).$$

Then for each element $x$ of $D_n$, it is known that the trace $\operatorname{tr}(\iota(x))$ and the determinant $\det(\iota(x))$ actually belongs to $K_n$ and that they do not actually depend on the choice of the splitting field $\Omega$.

We call them reduced trace and reduced norm of $x$ respectively.

As we already saw in the previous section, $S_n(k)$ is a splitting algebra of $A_n(k)$. Thus we see that the quotient field $L_n(k)$ of $S_n(k)$ is one of the splitting field of $D_n(k)$. We have thus proved that reduced norm and reduced determinant of $A_n(k)$ actually lie in $Z_n(k)$.