the results on totally positive units that we shall need.

It is worth pointing out that the third line of Eq. (\[newbound\]) is not necessarily an improvement over the bound in Prop. \[stollbnd\] if $\dim {{\rm Cl}}^+(L)[2]> \dim {{\rm Cl}}(L)[2]$. In our setting, we will seek conditions where $\dim {{\rm Cl}}^+(L)[2]= \dim {{\rm Cl}}(L)[2]$ and then our bound in Eq. (\[newbound\]) will be an improvement due to the more careful counting of units according to their infinite valuations.

Totally Positive Units {#sec-totally}
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Let $L$ be a totally real Galois number field of prime degree $p>2$, with embeddings $\tau_i : L \to {\mathbb R}$, for $i=1,\ldots,p$, and maximal order ${\mathcal{O}}_L$. Let ${{\rm Cl}}(L) = {{\rm Cl}}({\mathcal{O}}_L)$ be the ideal class group of $L$, and let ${{\rm Cl}}^+(L)$ be the narrow class group. Let $V_\infty = \{\pm 1\}^p\cong ({\mathbb F}_2)^p$ and, by abuse of notation, we extend the map ${{\rm res}}_\infty$ (as in Definition \[defn-res\], where we note that $H_\infty\cong V_\infty$ and $H\subseteq L^\times/(L^\times)^2$) to $${{\rm res}}_\infty: L^\times/(L^\times)^2 \to V_\infty$$ by ${{\rm res}}_\infty(\alpha) = ({{\rm sign}}(\tau_1(\alpha)),{{\rm sign}}(\tau_2(\alpha)),\ldots,{{\rm sign}}(\tau_p(\alpha)))$. Let ${\mathcal{O}}_L^\times$ be the unit group of ${\mathcal{O}}_L$, and let ${\mathcal{O}}_L^{\times, +}$ be the subgroup of totally positive units. Thus, $$\ker\left({{\rm res}}_\infty|_{{\mathcal{O}}_L^\times/({\mathcal{O}}_L^\times)^2}\right) = 