Corollary 3

Let $A^o$ be the opposite algebra of A. For every $a$ in A, we have

(40) Pcrd$_{A^o/K}(a; X) =$ Pcrd$_{A/K}(a; X)$,

(41) Trd$_{A^o/K}(a) =$ Trd$_{A/K}(a)$,

(42) Nrd$_{A^o/K}(a) =$ Nrd$_{A/K}(a)$.

Choose an extension L of K, a vector space V of dimension $n$ over L, and a homomorphism $\theta$ from A to End$_L(V)$ (VIII, p. 252, Theorem 1). Let $V^*$ be the dual vector space of V. The mapping that sends an element $a$ of A to the endomorphism $^t\theta (a)$ of $V^*$ is a K-algebra homomorphism from $A^o$ to End$_L(V^*)$. Corollary 3 then follows from Corollary 1 and Corollary 3 of III, §8, No. 4, p. 528.

The trace, norm, and characteristic polynomial of $a$ are therefore the same whether we view $a$ as an element of A or as an element of $A^o$. This property does not always hold when A is not assumed central simple (III, §9, p. 644, Exercise 1).