Class number one criterion for some non-normal totally real cubic fields

Let $\{K_m{\}}_{m\ge 4}$ be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial $f_m(x)=x^3-m{x}^2-(m+1)x-1$, where $m$ is an integer with $m\ge 4$. In this paper, we will give a class number one criterion for $K_m$.