Proposition 7

Let L be a maximal commutative semisimple subalgebra of A and $a$ an element of L. We have

(52) Pcrd$_{A/K}(a; X) =$ Pc$_{L/K}(a; X)$,

(53) Trd$_{A/K}(a) =$ Tr$_{L/K}(a)$,

(54) Nrd$_{A/K}(a) = N_{L/K}(a)$.

By Proposition 3 of VIII, p. 262, the L-modules A and $L^n$ are isomorphic; we therefore have the relation

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

Since the polynomial Pc$_{L/K}(a; X)$ is monic, it is therefore equal to the reduced characteristic polynomial Pcrd$_{A/K}(a; X)$ (VIII, p. 339, Lemma 2); this gives formula (52). By comparing the coefficients of $X^{n-1}$ (resp. the constant terms) on each side of (52), we obtain formula (53) (resp. (54)).