$79$, and $89$, so the corollary applies to all other primes not in this list (i.e., $3$, $5$, $11$, $13$, $17$, $19$, etc.).

\[thm-garbanati\] Let $K/{\mathbb Q}$ be an (imaginary) abelian extension of the rationals of degree $n$, let $h_K$ be the class number of $K$, and let $C_K\subseteq {\mathcal{O}}_K^\times$ the groups of circular units of $K$ (as defined in [@garbanati], p. 376), and let $C_K^+ \subseteq {\mathcal{O}}_K^{\times, +}$ be the subgroup of circular units that are totally positive. Let $K^+$ be the maximal real subfield of $K$, and let $h_K^+$ be its class number. Let $h_K^- = h_K/h_K^+$. Further, assume that each prime $p$ which ramifies in $K$ does not split in $K^+$. Then:

1.  The index $[{\mathcal{O}}_K^{\times,+}:({\mathcal{O}}_K^\times)^2] $ is a divisor of the index $[C_K^{+}:C_K^2]$.

2.  If the discriminant of $K$ is plus or minus a power of a prime, then $h_K^-$ is odd if and only if $[C_L^{+}:C_L^2]=1$, where $C_L$ is the subgroup of circular units of $L=K^+$.

3.  Suppose the discriminant of $K$ is plus or minus a power of a prime, $[K^+:{\mathbb Q}]=n$ is a power of an odd prime $p$, and the order of $2\bmod p$ is even. Then, $h_K^-$ is odd if and 