Corollary 2

*Let A be a central simple algebra of finite degree over K. Let B be a semisimple K-algebra, let f and g be algebra homomorphisms from B to A, and let $\mathscr{B}$ be a generating subset of the K-vector space B. The following properties are equivalent:

(i) There exists an inner automorphism $\theta$ of A such that $g = \theta \circ f$. (ii) For every $b \in \mathscr{B}$, we have $\mathrm{Pc}_{A/K}(f(b); X) = \mathrm{Pc}_{A/K}(g(b); X)$.* When K has characteristic zero, these properties are equivalent to the following:

(iii) For every $b\in \mathscr{B}$, we have Tr$_{A/K}(f(b)) =$ Tr$_{A/K}(g(b))$.

Denote by M and N the left B-modules obtained by endowing the additive group of A with the external laws $(b, a)\mapsto f(b)a$ and $(b, a)\mapsto g(b)a$, respectively. By Proposition 1 of VIII, p. 257, property (i) is equivalent to the fact that the B-modules M and N are isomorphic. By construction, the homothety $b_M$ associated with an element $b$ of B is left multiplication by $f(b)$ in A; consequently, we have the relations

Pc$_M(b; X) =$ Pc$_{A/K}(f(b); X)$,

Tr$_M(b) =$ Tr$_{A/K}(f(b))$

and two analogous relations where we replace M with N and $f$ with $g$. We know (VIII, p. 386, Corollary 1) that the B-modules M and N are isomorphic if and only if we have Pc$_M(b; X) =$ Pc$_N(b; X)$ for every $b\in \mathscr{B}$. When the field K has characteristic 0, this relation is also equivalent to Tr$_M(b) =$ Tr$_N(b)$ for every $b\in \mathscr{B}$ (VIII, p. 384, Corollary of Proposition 6). The equivalence of properties (i) and (ii), and also of (i) and (iii) when K has characteristic 0, follows.