Remark 20

Suppose that the conditions of No. 3 are satisfied and in addition that $A$ is equal to $K$ or to $K(i)$. Apply the results of Exercises 17 to 19 to the extension of $\Phi$ to the $p$-th exterior powers (§ 1, No. 9) and to the $p$-th exterior powers of the endomorphisms or matrices considered. In particular, now prove that, if $\prod_{i=1}^n (X - \lambda_i)$ and $\prod_{i=1}^n (X - \rho_i^2)$ are the decompositions into linear factors of the characteristic polynomials of an endomorphism $u$ of $E$ and of the endomorphism $u^*u$, and if one supposes $|\lambda_i| \geqslant |\lambda_{i+1}|$ and $\rho_i \geqslant \rho_{i+1} \geqslant 0$ for $1 \leqslant i \leqslant n-1$, one has

for $1 \leqslant h \leqslant n-1$ and $|\lambda_1 \lambda_2 \ldots \lambda_n| = \rho_1 \rho_2 \ldots \rho_n$.