Determination of all imaginary cyclic quartic fields of prime class number p ≡ 3(mod4), and non-divisibility of class numbers

Abstract

Let [Formula: see text] be a prime such that [Formula: see text]. Then, we show that there is no imaginary cyclic quartic extension [Formula: see text] of [Formula: see text] whose class number is [Formula: see text]. Suppose [Formula: see text] is a cyclic extension of number fields with an odd degree. Then, we show that [Formula: see text] does not divide the class number of [Formula: see text] if the class group of [Formula: see text] is cyclic. We also construct some families of number fields whose class number is not divisible by a fixed prime.