- Tytuł:
- Solutions of cubic equations in quadratic fields
- Autorzy:
-
Chakraborty, K.
Kulkarni, Manisha - Powiązania:
- https://bibliotekanauki.pl/articles/1390522.pdf
- Data publikacji:
- 1999
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
elliptic curves
Diophantine equation - Opis:
- 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].
- Źródło:
-
Acta Arithmetica; 1999, 89, 1; 37-43
0065-1036 - Pojawia się w:
- Acta Arithmetica
- Dostawca treści:
- Biblioteka Nauki
Artykuł