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Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$.

Question: Is the class number of $A_n$ always equal to one, or equivalently, is the ring of integers of $A_n$ a principal ideal domain?

This is true for $n \leq 16$ using MAGMA and also true for $n \leq 30$ assuming the GRH (according to MAGMA).

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    $\begingroup$ Just a side remark: These fields have been studied, e.g. Bary-Soroker, Shusterman and Zannier in the Appendix to arxiv.org/abs/1508.05363 prove that their maximal totally real subfield is $\mathbb{Q}$. I am not aware of results regarding their class number though. $\endgroup$
    – Arno Fehm
    Jun 17, 2020 at 6:54

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