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).