Solutions of cubic equations in quadratic fields

Chakraborty, K and Kulkarni, Manisha V (1999) Solutions of cubic equations in quadratic fields. In: Acta Arithmetica, 89 (1). 37-43 .

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Abstract

Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r + s + t = rst = 1 in O-K. This Diophantine equation gives an elliptic curve defined over Q with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of Q(i) and Q(root 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

Item Type:	Journal Article
Publication:	Acta Arithmetica
Publisher:	Institute of Mathematics of the Polish Academy of Sciences
Additional Information:	Copyright of this article belongs to Institute of Mathematics of the Polish Academy of Sciences.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	02 Jul 2011 06:25
Last Modified:	02 Jul 2011 06:25
URI:	http://eprints.iisc.ac.in/id/eprint/38769

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