Rational Points on elliptic curves

**ABSTRACT:
**This presentation will cover the basic geometries and algebra
of cubics as they pertain to rational points, and a little background on
certain topics. I will establish that the set of rational solutions to a cubic
equation form a group, which is abelian. The main focus will be to give a brief
proof of Mordell's Theorem. **Mordell's Theorem** - If a non-singular
plane cubic curve has a rational point, then the group of rational points is
finitely generated.

I will prove some of the lemmas necessary to prove the theorem but not all of them will be proven. If time allows, concrete examples will be given.