Prime factors of class number of cyclotomic fields (Q1011967)

Let \(p\) be an odd prime, \(\zeta\) a primitive \(p\)-th root of unity, and \(K = \mathbb Q(\zeta)\) the cyclotomic field defined by \(\zeta\). Kummer's class number formula for the minus class number \(h^-(K)\) involves a product of values of the polynomial \(F(x) = \sum_{k=0}^{p-2} r_{-k}x^k\), where \(r_k\) is the smallest positive residue of \(r^k \bmod p\). Roland Quême asked whether the \(\ell\)-rank of the class group of \(K\) (for primes \(\ell \neq p\)) is equal to the degree of \(\gcd(F(x),x^N+1)\), where \(N = (p-1)/2\). Here the author shows that \(\ell = 3\) and \(p = 3299\) provides a counterexample: already the quadratic subfield \(\mathbb Q(\sqrt{-p}\,)\) of \(K\) has \(3\)-rank \(2\), whereas \(\gcd(F(x),x^N+1) = x+1\). The weaker questions whether the degree of \(\gcd(F(x),x^N+1)\) is a lower bound for the \(\ell\)-rank of the class group, or whether it is equal to the \(\ell\)-rank of a suitably chosen relative class group, remain open.