$q=2p+1$, let $L={\mathbb Q}(\zeta_q+\zeta_q^{-1})$, where $\zeta_q$ is a primitive $q$-th root of unity, and let $K={\mathbb Q}(\zeta_q)$. Then, $C^+_L=C_L^2$ if and only if the class number of $K$ is odd.

If $h^+_K$ is even, then $h^-_K$ is even (see, for instance, [@stevenhagen], pp. 773-774, for a proof of this fact). Since $h_K=h_K^+h_K^-$, it follows that $h_K^-$ is odd if and only if $h_K$ is odd. Since the discriminant of $K$ is a power of $q$ (prime), Theorem \[thm-garbanati\], part (2), implies that $h^-_K$ is odd if and only if $C^+_L = (C_L)^2$. Hence, the Davis–Taussky conjecture holds if and only if $h_K^-$ is odd, if and only if $h_K$ is odd.

We conclude this section with some remarks about how to compute an upper bound of $\rho_\infty$ in the cyclic case, working in coordinates over ${\mathbb F}_2$. Let $L$ be a cyclic extension of ${\mathbb Q}$ of degree $p>2$, and let ${\operatorname{Gal}}(L/{\mathbb Q})=\langle \sigma \rangle$. Then, $L$ is totally real (since $L$ is Galois over ${\mathbb Q}$, then it is totally real or totally imaginary, so $[L:{\mathbb Q}]=p>2$ is odd, and $p=2r_2$ is impossible). Let $u\neq \pm 1$ be a fixed (known) unit in ${\mathcal{O}}_L^\times$ and let $${\rm res}_\infty(u) = 