Solutions of cubic equations in quadratic fields
K. Chakraborty ; Manisha V. Kulkarni
Acta Arithmetica, Tome 89 (1999), p. 37-43 / Harvested from The Polish Digital Mathematics Library
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Let K be any quadratic field with 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 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(√2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

Publié le : 1999-01-01
EUDML-ID : urn:eudml:doc:207257 
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     title = {Solutions of cubic equations in quadratic fields},
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     volume = {89},
     year = {1999},
     pages = {37-43},
     zbl = {0935.11019},
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K. Chakraborty; Manisha V. Kulkarni. Solutions of cubic equations in quadratic fields. Acta Arithmetica, Tome 89 (1999) pp. 37-43. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav89i1p37bwm/

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