Corollary 1

Let L be an extension of K. Let V be a vector space of dimension $n$ over L and $\theta$ a K-algebra homomorphism from A to End$_L(V)$. For every element $a$ of A, we have

(31) Pcrd$_{A/K}(a; X) =\chi_{\theta(a)}(X)$,

(32) Trd$_{A/K}(a) =$ Tr($\theta (a)$),

(33) Nrd$_{A/K}(a) =$ det($\theta (a)$).

Let $\theta '$ be the L-algebra homomorphism from $A_{(L)}$ to End$_L(V)$ such that $\theta '(\lambda \otimes a) =\lambda \theta (a)$ for $\lambda \in L$ and $a\in A$. The algebra $A_{(L)}$ is simple by Corollary 2 of VIII, p. 222; the homomorphism $\theta '$ is therefore injective. But the algebras $A_{(L)}$ and End$_L(V)$ over the field L have the same degree, equal to $n^2$, so $\theta '$ is an isomorphism. Corollary 1 then follows from Proposition 4 and Example 2 above.