Corollary 2

Let $a$ and $a'$ be elements of A and $\lambda$ an element of K. We have the relations

(34) Pcrd$_{A/K}(a;a) = 0$,

(35) Pcrd$_{A/K}(\lambda a;\lambda X) =\lambda^n$ Pcrd$_{A/K}(a; X)$,

(36) Trd$_{A/K}(a+a') =$ Trd$_{A/K}(a) +$ Trd$_{A/K}(a')$,

Trd$_{A/K}(\lambda a) =\lambda$ Trd$_{A/K}(a)$,

(37) Trd$_{A/K}(aa') =$ Trd$_{A/K}(a'a)$,

(38) Nrd$_{A/K}(aa') =$ Nrd$_{A/K}(a)\cdot$ Nrd$_{A/K}(a')$,

Nrd$_{A/K}(\lambda a) =\lambda^n$ Nrd$_{A/K}(a)$,

(39) Trd$_{A/K}(1) =n$, Nrd$_{A/K}(1) = 1$.

Since A is central simple and has reduced degree $n$ over K, there exist an extension L of K and a vector space V of dimension $n$ over L such that $A_{(L)}$ is isomorphic to the algebra End$_L(V)$ (VIII, p. 252, Theorem 1). Corollary 2 then follows from Corollary 1 and the properties of the trace and determinant of an endomorphism. In particular, formula (34) follows from the Cayley– Hamilton theorem (III, §8, No. 11, p. 541, Proposition 20, cf. also VIII, p. 338, Remark 2).