CM-fields with relative class number one

Abstract

We will show that the normal CM-fields with relative class number one are of degrees <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to 216"> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>216</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leq 216</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to 96"> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>96</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leq 96</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the CM-fields with class number one are of degrees <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to 104"> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>104</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leq 104</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By many authors all normal CM-fields of degrees <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to 96"> <mml:semantics> <mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>96</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\leq 96</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with class number one are known except for the possible fields of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="64"> <mml:semantics> <mml:mn>64</mml:mn> <mml:annotation encoding="application/x-tex">64</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="96"> <mml:semantics> <mml:mn>96</mml:mn> <mml:annotation encoding="application/x-tex">96</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Consequently the class number one problem for normal CM-fields is solved under the Generalized Riemann Hypothesis except for these two cases.