only if $h_K^+$ is odd.

The results are, respectively, Lemma 5, Theorem 3, and Theorem 4 of [@garbanati].

Next, we cite a results of Estes which extends work of Davis ([@davis Corollary 2]), and Stevenhagen ([@stevenhagen Theorem 2.5]). See also [@kim Theorems 3.1 and 3.3].

\[thm-estes\] Let $q$ and $p$ be primes such that $q=2p+1$. If $2$ is inert in ${\mathbb Q}(\zeta_p)^+$, where $\zeta_p$ is a primitive $p$-th root of unity, then the class number of ${\mathbb Q}(\zeta_q)$ is odd.

The following result combines the results of Davis, Estes, Stevenhagen, and Garbanati, and gives a specific criterion to check that $\rho_\infty=0$ for the maximal real subfield of a cyclotomic field.

\[thm-kim\] Let $q$ and $p>2$ be primes such that $q=2p+1$, and let $L={\mathbb Q}(\zeta_q+\zeta_q^{-1})$, where $\zeta_q$ is a primitive $q$-th root of unity. Further, assume that the prime $2$ is inert in the extension ${\mathbb Q}(\zeta_p)^+/{\mathbb Q}$. Then, $\rho_\infty = \dim_{{\mathbb F}_2} {\mathcal{O}}_L^{\times,+}/({\mathcal{O}}_L^\times)^2 = 0.$

Let $K={\mathbb Q}(\zeta_q)$ and let $K^+ = L = {\mathbb Q}(\zeta_q+\zeta_q^{-1})$. We note that the discriminant of $K$ is a power of a prime (namely $q$), and therefore the primes of $K$ or $L$ that are ramified (namely the primes above $q$), are totally ramified, 