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In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is 10921 = 67 • 163 , which is the conductor of the biquadratic number field Q(\/-67, v'-163). _ *=(Xl}X"-x(Xr},doi:10.3792/pjaa.68.21 fatcat:yzj5nh7dvbha3e5mspufyfenya