What is the Galois group of narrow Hilbert class field over Hilbert class field?

I am trying to solve the following question:

Let $K$ be a number field, $H'$ and $H$ be the narrow Hilbert class field and Hilbert class field respectively. Let $O^*_{K,+}\subset O^*_{K}$ be the group of totally positive units. Show that $Gal(H'/H)\cong (Z/2Z)^{r-t}$ where $r$ is the set of real places and $2^t=[ O^*_{K}:O^*_{K,+}]$.

Here is my effort: By the main theorem of CFT, we know that $Gal(H'/H)$ is exactly the kernel of the canonical surjective morphism $f: Cl(K, O_K)\to Cl(K)$, in order to compute the kernel, I use the Propostion 6.114 of Kazuya Kato's book, which is as follows: enter image description here

Since $R^*/R^*_{>0}\cong Z/2Z$, we just need to compute how many real place $\sigma$ of $K$ such that $[O^*_{\sigma{(K)}}:O^*_{\sigma{(K)},+}]=2$, we denote the number as $d$, then we need to show $d=t$. But I have no idea how to show it.

PS. My text book about CFT is Kazuya Kato's book, he define it without modulus, so I hope we can discuss it without using the concept of modulus.

Solution 1:

$\require{AMScd}$ Kato's $Cl(K,O_K)$ is usually denoted by $Cl^+(K)$. These groups are defined by the following diagrams, where $I_K$ is the group of fractional ideals, $P_K$ that of principal ideals, and $P_K^+$ that of totally positive ideals: \begin{CD} 1 @>>> P_K^+ @>>> I_K @>>> Cl^+(K) @>>> 1 \\ @. @VVV @VVV @VVV @. \\ 1 @>>> P_K^ @>>> I_K @>>> Cl(K) @>>> 1 \end{CD} By the snake lemma, the group you're interested in is isomorphic to $P_K/P_K^+$.

The commutative diagram \begin{CD} 1 @>>> O_{K,+}^* @>>> K^\times_+ @>>> P_K^+ @>>> 1 \\ @. @VVV @VVV @VVV @. \\ 1 @>>> O_K^* @>>> K^\times @>>> P_K @>>> 1 \end{CD} now answers your question since the quotient in the middle is $K^\times/K^{\times}_+ \simeq ({\mathbb Z}/2{\mathbb Z})^r$