The simplest cubic fields

Class Numbers of the Simplest Cubic Fields

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 ...

Integral Points on Elliptic Curves Defined by Simplest Cubic Fields

CONTENTS Introduction 1. Elliptic Curves Defined by Simplest Cubic Fields 2. Linear Forms in Elliptic Logarithms 3. Computation of Integral Points 4. Tables of Results 5. General Results about Integral Points on the Elliptic Curves y2 = x3 + mx2 (m+3)x + 1 References Let f(X) be a cubic polynomial defining a simplest cubic field in the sense of Shanks. We study integral points on elliptic curve...

On Correspondence between Solutions of a Parametric Family of Cubic Thue Equations and Non-isomorphic Simplest Cubic Fields

We give a correspondence between non-trivial solutions to a parametric family of cubic Thue equations X − mXY − (m + 3)XY 2 − Y 3 = k where k | m + 3m+ 9 and non-isomorphic simplest cubic fields. By applying R. Okazaki’s result for non-isomorphic simplest cubic fields, we obtain all solutions to the family of cubic Thue equations for k | m + 3m+ 9.

On Correspondence between Solutions of a Parametric Family of Cubic Thue Equations and Isomorphic Simplest Cubic Fields

We give a correspondence between non-trivial solutions to a parametric family of cubic Thue equations X − mXY − (m + 3)XY 2 − Y 3 = k where k | m+3m+9 and isomorphic simplest cubic fields. By applying R. Okazaki’s result for isomorphic simplest cubic fields, we obtain all solutions to the family of cubic Thue equations for k | m + 3m + 9.

Power Integral Bases in the Family of Simplest Quartic Fields