Quadratic fields with 3-rank equal to 4

Fredéric Diaz, Daniel Shanks, Hywel C Williams

Type: Article

Publication Date: 1979-01-01

Citations: 10

DOI: https://doi.org/10.1090/s0025-5718-1979-0521299-4

Abstract

In [2] there is reference to 119 known imaginary quadratic fields that have 3-rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than-or-slanted-equals 4"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r \geqslant 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We examine these fields and determine the exact values of <italic>r</italic>. Their associated real fields and the distribution of their 3-Sylow subgroups are also studied. Some of the class groups are recorded since they are of special interest. These include examples having an infinite class field tower and only one ramified prime, and others having an infinite tower because of two different components of their class groups.

Locations

Mathematics of Computation - View - PDF

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