Some analytic estimates of class numbers and discriminants

Abstract

This paper presents some new lower bounds (Theorem 1) for the absolute value of the discriminant of a number field. Although not as good for large degrees as some of the author's recent estimates, these bounds are easy to prove and are applicable even for very small degrees. Our main application of these bounds is to class numbers of totally complex quadratic extensions of totally real number fields. Let X denote the set of number fields k for which there exists a sequence of fields Q = ko c . . . c kt = k, each normal over the preceding one. Then for any fixed h there exist only finitely many totally complex number fields K with class number equal to h, which contain totally real subfields k such that [ ,K:k]=2 and ke~#. Moreover, given h, all such K with [,K:Q] >6 can be determined effectively. (The effective computability is also known for [ , K : Q ] 6. If the Generalized Riemann Hypothesis is true, then we obtain an effective result even if we drop both of the requirements k e ~ and [ K : Q ] > 6 . Furthermore as an easy corollary of the above result we also obtain information about class numbers of totally complex abelian extensions of totally real number fields.