On the divisibility of class numbers in families of number fields.

Abstract

We adapt techniques used to investigate the divisibility of class numbers in families of algebraic number fields to study related topics in three particular families of number fields. Let $r$ be a positive integer. We prove that the quartic family contains infinitely many distinct fields whose class group contains a cyclic subgroup of order $r$. We further show that when $r$ is odd, an ideal class generating this subgroup does not come from the field's unique quadratic subfield. When $r$ is odd, we show that the family of sextics contains infinitely many distinct fields whose class group contains a cyclic subgroup of order $r$ generated by an ideal class which does not come from either the quadratic or cubic subfields of the sextic field. Finally, we extend and modify the techniques to handle a family of non-Galois cubic extensions of $\mathbb{Q}.$ We prove that here too the result holds; this family contains infinitely many distinct fields whose class group contains a cyclic subgroup of order $r.$