  Let $X\subset \P^n$ be a possibly singular hypersurface of degree $d\le n$,
defined over a finite field $k$. We show that the diagonal, suitably
interpreted, is decomposable. This gives a proof that the eigenvalues of the
Frobenius action on its $\ell$-adic cohomology $H^i(\bar{X}, \Q_\ell)$, for
$\ell \neq {\rm char}(k)$, are divisible by $q$, without using the result on
the existence of rational points by Ax and Katz.
