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

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