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I am interested in finding out when the narrow class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ is the same as the class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ where $\zeta_p$ is a primitive $p$th root of unity and $p$ a prime.

Any help would be much appreciated.

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  • $\begingroup$ keywords: signature cyclotomic units $\endgroup$
    – Franz Lemmermeyer
    Commented Dec 7, 2022 at 22:51
  • $\begingroup$ Franz is of course correct, and for info, for $p<163$ the class number is $1$ (well-known) and the narrow class number is also $1$ except for $p=29$ and $p=113$ for which it is equal to $8$. For $p=163$ they are both equal to $4$. $\endgroup$
    – Henri Cohen
    Commented Dec 7, 2022 at 23:22
  • $\begingroup$ Thank you both. @HenriCohen do you have a precise reference for this (the narrow class number)? $\endgroup$
    – did
    Commented Dec 8, 2022 at 17:04
  • $\begingroup$ Among the references given in the preface of link.springer.com/book/10.1007/978-3-030-01512-1, there is Verhoek's thesis universiteitleiden.nl/binaries/content/assets/science/mi/… $\endgroup$
    – Franz Lemmermeyer
    Commented Dec 8, 2022 at 17:36

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