Example 1

By Theorem 1 of VIII, p. 120, there exist an integer $r\geqslant 1$ and a field D such that A is isomorphic to $\mathbf{M}_r(D)$. Let $d$ be the reduced degree of D over K; we have $r=n/d$. Let M be an A-module of finite length $\ell$; we will prove the formula

(26) Pc$_{M/K}(a_M; X) =$ Pcrd$_{A/K}(a; X)^{d\ell}$

for every element $a$ of A. The A-module $A_s$ has length $r$ (VIII, p. 121, Lemma 2). The A-modules $M^r$ and $A^{\ell}_s$ have the same length, so they are isomorphic, and we have

Pc$_{M/K}(a_M; X)^r=$ Pc$_{A/K}(a; X)^{\ell}$

by formula (15) of III, §9, No. 2, p. 542. Since we have $rd=n$, formula (26) follows from formula (20) and Lemma 2 (VIII, p. 339).