Non-normal class number one problem and the least prime power-residue (Q2707575)

Let \(K\) be a totally complex quadratic extension of a cyclic cubic number field of conductor \(f\), and assume that \(K/\mathbb Q\) is not normal. Then the quotient of class numbers \(h(K/F) = h(K)/h(F)\) is an integer called the relative class number, and the discriminant \(d_K\) of \(K\) is divisible by the square of \(d_F = f^2\), so \(d = d_K/f^4\) is an integer. Let \(\chi\) denote the cubic character associated to \(K/\mathbb Q\). The following inequalities from the author's theorem 2 show that if primes with certain properties exist, then we must have \(h(K/F) > 1\). Let \(l\) be a prime and assume that \(h(K/F) = 1\); NEWLINENEWLINENEWLINEa) if \(\chi(l) \neq 1\) and \((d/l) = -1\), then \(l > f/18 + |d|^{1/3}/4\); NEWLINENEWLINENEWLINEb) if \(\chi(l) = 1\) and \((d/l) \neq -1\), then \(l^2 > f/18 + |d|^{1/3}/4\). NEWLINENEWLINENEWLINEThe author proves a similar but simpler result for quartic CM fields. Results of this type are very useful for solving class number \(1\) problems for CM-fields.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].