Rational Distances to the Corners of the Unit Square

Abstract

The objective of this thesis is to create a concrete statement about the existence of rational points in the unit square. We will use the following definition for a rational point: A rational point is defined as a point whose distances to the vertices of a geometric figure are all rational. Notice that a number theoretic rational point is a point (x, y) in an algebraic curve f(x, y) = 0 where x and y are rational. We will present a variety of methods to prove certain properties of these rational points. Finally we prove an equivalence between the non-existence of rational points on the edges of the unit square and the absence of integer roots for certain families of polynomials.