Definition 1

Let $a$ be an element of the algebra A. The reduced characteristic polynomial of $a$ (with respect to A) is the unique monic polynomial in K[X], denoted by Pcrd$_{A/K}(a; X)$, that satisfies the relation

(20) Pc$_{A/K}(a; X) =$ Pcrd$_{A/K}(a; X)^n$.

Let $a$ be an element of A. Since A has degree $n^2$ over K, the polynomial Pc$_{A/K}(a; X)$ has degree $n^2$, so Pcrd$_{A/K}(a; X)$ is a monic polynomial of degree $n$; let us write it as

(21) Pcrd$_{A/K}(a; X) = X^n+\sum^{n-1}_{r=0}(-1)^rb_r(a)X^{n-r}$.

We set

(22) Trd$_{A/K}(a) =b_1(a)$, Nrd$_{A/K}(a) =b_n(a)$.