Solutions of cubic equations in quadratic fields

Let K be any quadratic field with O-K its ring of integers. We study the so lutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solut ions by using elementary tools. As an application we consider the Diophanti ne 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 stu dy of the solutions of cubic equations in quadratic fields we present a sim ple proof of the fact that except for the ring of integers of Q(i) and Q(ro ot 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].