Divisibility of class numbers of non-normal totally real cubic number fields

Abstract

In this paper, we consider a family of cubic fields {K-m}(m >= 4) associated to the irreducible cubic polynomials P-m(x) = x(3) - mx(2) - (m+1)x - 1, (m >= 4). We prove that there are infinitely many {K-m}(m >= 4)'s whose class numbers are divisible by a given integer n. From this, we find that there are infinitely many non-normal totally real cubic fields with class number divisible by any given integer n.