Salem numbers of negative trace

We prove that, for all $d\geq 4$, there are Salem numbers of degree $2d$ and trace $-1$, and that the number of such Salem numbers is $\gg d/\left( \log \log d\right) ^{2}$. As a consequence, it follows that the number of totally positive algebraic integers of degree $d$ and trace $2d-1$ is also $\gg d/\left( \log \log d\right) ^{2}$.