The determination of the imaginary abelian number fields with class number one

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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 is10921=67⋅16310921 = 67 \cdot 163, which is the conductor of the biquadratic number fieldQ(−67,−163){\mathbf {Q}}(\sqrt { - 67} ,\sqrt { - 163} ).