Solutions of cubic equations in quadratic fields

• Autorzy: K. Chakraborty , Manisha V. Kulkarni
• Języki publikacji: EN

Abstrakty

EN

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(√2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

Słowa kluczowe

EN

elliptic curves; Diophantine equation

Kategorie tematyczne

• 11D25: Cubic and quartic equations
• 11D41: Higher degree equations; Fermat's equation
• 11G05: Elliptic curves over global fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1999
• Tom: 89
• Numer: 1
• Strony: 37-43

Daty

wydano

1999

otrzymano

1997-09-23

poprawiono

1998-09-29

Twórcy

autor

K. Chakraborty

• Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India

autor

Manisha V. Kulkarni

• Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Bibliografia

• [1] J. W. S. Cassels, On a diophantine equation, Acta Arith. 6 (1960), 47-52.
• [2] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge Univ. Press, Cambridge, 1992.
• [3] M. Laska, Solving the equation x³ - y² = r in number fields, J. Reine Angew. Math. 333 (1981), 73-85.
• [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.
• [5] G. Sansone et J. W. S. Cassels, Sur le problème de M. Werner Mnich, Acta Arith. 7 (1962), 187-190.
• [6] W. Sierpiński, Remarques sur le travail de M. J. W. S. Cassels 'On a diophantine equation', Acta Arith. 6 (1961), 469-471.
• [7] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, 1986.
• [8] C. Small, On the equation xyz = x+y+z = 1, Amer. Math. Monthly 89 (1982), 736-749.