On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Details On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

2008-10-20

arxiv.org/abs/0810.3374v4

Mathematics

Number Theory

15 pages, to appear in J. Number Theory

Scientific paper

Let $m\geq -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations \[ X^3-mX^2Y-(m+3)XY^2-Y^3=\lambda \] where $\lambda>0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $\lambda$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389$.

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