A lower bound for the class number of certain cubic number fields

Abstract

Let K be a cyclic number field with generating polynomial \[ X 3 − a − 3 2 X 2 − a + 3 2 X − 1 {X^3} - \frac {{a - 3}}{2}{X^2} - \frac {{a + 3}}{2}X - 1 \] and conductor m. We will derive a lower bound for the class number of these fields and list all such fields with prime conductor m = ( a 2 + 27 ) / 4 m = ({a^2} + 27)/4 or m = ( 1 + 27 b 2 ) / 4 m = (1 + 27{b^2})/4 and small class number.