Class number and class group problems for some non-normal totally real cubic number fields

Abstract:

Let \lcub;K _m } _{m ≥ 4} be the family of non-normal totally real cubic number fields associated with the Q-irreducible cubic polynomials P _m (x) =x ³−mx ²−(m+1)x− 1, m≥ 4. We determine all these K _m 's with class numbers h _m ≤ 3: there are 14 such K _m 's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents e _m of the ideal class groups of these K _m go to infinity with m and we determine all these K _m 's with ideal class groups of exponents e _m ≤ 3: there are 6 suchK _m with ideal class groups of exponent 2, and 6 such K _m with ideal class groups of exponent 3.