Corollary 1

Let E and F be semisimple A-modules of finite dimension over K, and let $\mathscr{A}$ be a generating subset of the K-vector space A. Suppose that for every $a \in \mathscr{A}$, the characteristic polynomials of the endomorphisms $a_E$ and $a_F$ of the K-vector spaces E and F are equal. Then the A-modules E and F are isomorphic.

Let $a$ be an element of $\mathscr{A}$. The characteristic polynomials of $a_E$ and $a_F$ have the same degree, so the dimension of E is equal to that of F; we denote it by $n$. Let $\mathrm{Pc}_E(a; T)$ be the characteristic polynomial of $a_E$. In $K(T)$, we have the equalities

and $\chi_F(a; T)$ is given by an analogous formula. Because of our assumptions, we have $\chi_E(a; T) = \chi_F(a; T)$. By Theorem 2, we have [E] = [F], which implies that E and F are isomorphic (VIII, p. 190, Corollary of Proposition 7).