Let $K=\mathbb Q(\xi\_{39})$ be the 39-th cyclotomic field. Pari-GP told me that the prime ideals above $3$ and $13$ are not principal. Is there a way to prove that by hand (no computation made by computer)



Non-principal ideals in cyclotomic fields
The quadratic number field $k = {\mathbb Q}(\sqrt{-39})$ has a cyclic class group of order 
$4$. Its genus field is $K = {\mathbb Q}(\sqrt{-3},\sqrt{13})$, which has class number $2$.
The Hilbert class field of $K$ is dihedral over ${\mathbb Q}$, hence not contained in the
cyclotomic field $L = {\mathbb Q}(\zeta\_{39})$. This already shows that the class number of $L$ is even. 


Now verify that one of the prime ideals above $13$ in $K$ is not principal (obvious since 
its norm down to $k$ is not principal) and that it ramifies completely in $L/K$. 


