Solutions of cubic equations in quadratic fields

K. Chakraborty; Manisha V. Kulkarni

Acta Arithmetica (1999)

  • Volume: 89, Issue: 1, page 37-43
  • ISSN: 0065-1036

Abstract

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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].

How to cite

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K. Chakraborty, and Manisha V. Kulkarni. "Solutions of cubic equations in quadratic fields." Acta Arithmetica 89.1 (1999): 37-43. <http://eudml.org/doc/207257>.

@article{K1999,
abstract = {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].},
author = {K. Chakraborty, Manisha V. Kulkarni},
journal = {Acta Arithmetica},
keywords = {elliptic curves; Diophantine equation; quadratic field; cubic diophantine equations},
language = {eng},
number = {1},
pages = {37-43},
title = {Solutions of cubic equations in quadratic fields},
url = {http://eudml.org/doc/207257},
volume = {89},
year = {1999},
}

TY - JOUR
AU - K. Chakraborty
AU - Manisha V. Kulkarni
TI - Solutions of cubic equations in quadratic fields
JO - Acta Arithmetica
PY - 1999
VL - 89
IS - 1
SP - 37
EP - 43
AB - 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].
LA - eng
KW - elliptic curves; Diophantine equation; quadratic field; cubic diophantine equations
UR - http://eudml.org/doc/207257
ER -

References

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  1. [1] J. W. S. Cassels, On a diophantine equation, Acta Arith. 6 (1960), 47-52. Zbl0094.25701
  2. [2] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge Univ. Press, Cambridge, 1992. Zbl0758.14042
  3. [3] M. Laska, Solving the equation x³ - y² = r in number fields, J. Reine Angew. Math. 333 (1981), 73-85. 
  4. [4] R. A. Mollin, C. Small, K. Varadarajan and P. G. Walsh, On unit solutions of the equation xyz = x+y+z in the ring of integers of a quadratic field, Acta Arith. 48 (1987), 341-345. Zbl0576.10009
  5. [5] G. Sansone et J. W. S. Cassels, Sur le problème de M. Werner Mnich, Acta Arith. 7 (1962), 187-190. 
  6. [6] W. Sierpiński, Remarques sur le travail de M. J. W. S. Cassels 'On a diophantine equation', Acta Arith. 6 (1961), 469-471. 
  7. [7] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, 1986. 
  8. [8] C. Small, On the equation xyz = x+y+z = 1, Amer. Math. Monthly 89 (1982), 736-749. Zbl0505.12024

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