Class numbers of the simplest cubic fields

Washington, Lawrence C.

doi:10.1090/S0025-5718-1987-0866122-8

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Class numbers of the simplest cubic fields

Author: Lawrence C. Washington
Journal: Math. Comp. 48 (1987), 371-384
MSC: Primary 11R16; Secondary 11R20, 14K07
MathSciNet review: 866122
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Abstract: Using the "simplest cubic fields" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by n, for any n. Using 2-descents on elliptic curves, we obtain precise information on the 2-Sylow subgroups of the class groups of these fields. A theorem of H. Heilbronn associates a set of quartic fields to the class group. We show how to obtain these fields via these elliptic curves.

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