On class numbers of quadratic extensions of algebraic number fields (Q1112109)

Let \(t>1\) be any integer. It is proved that for any algebraic number field having a totally ramified rational odd prime p there are infinitely many quadratic extensions L of K such that \(t| h(L)\). Furthermore, L may be chosen to be of the form K(\(\sqrt{n})\) where n is any square-free rational integer of the form \(r^ 2-m^ 2\) so that p does not divide n and r is restricted by the equation \(r^ 2\leq m^{t-1}(m-1)\).