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This is an excerpt from the paper "Class groups, totally positive units, and squares" (page 36). I am struggling to understand the last equality $|K^{(1)}_{2}:K|=|\overline{O}_K^{+}|$, the bar convention means modulo squares. In particular, I do not see why $|\overline{O}_K^{+}|$ should be finite.

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  • $\begingroup$ Surely that O should be a U. $\endgroup$
    – Dror Speiser
    Commented Dec 12, 2018 at 21:49
  • $\begingroup$ That's my thoughts too, especially if you consider the statement that $O^+_K$ is a subgroup. However, $U_K$ was considered in the paper previously, and just below in the proof it states: "By the self-duality established above, we may use exactly the same reasoning as used by Taylor on $\overline{U}^+_K$ and $\overline{U}^0_K$ in [11, (*), p. 157] to establish $\overline{O}^+_K\cong \overline{O}^0_K$". $\endgroup$
    – pavl0
    Commented Dec 12, 2018 at 22:50

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