In the statement of the following theorem, a nontrivial solution to a homogeneous polynomial equation is a solution where not all indeterminates are 0.
The analogue of Theorem 14.2.14 for cubic equations is false. For example, Selmer proved that the cubic
Open Problem.
Give an algorithm that decides whether or not a cubic
This open problem is closely related to the Birch and Swinnerton-Dyer Conjecture for elliptic curves. The truth of the conjecture would follow if we knew that ``Shafarevich-Tate Groups'' of certain elliptic curves are finite.
William Stein 2012-09-24