Geometry and Arithmetic

• Joseph H. Silverman
• John Tate
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

Everyone knows what a rational number is, a quotient of two integers. We call a point in the (x, y) plane a rational point if both its coordinates are rational numbers. We call a line a rational line if the equation of the line can be written with rational numbers; that is, if its equation is
$$ax + by + c = 0$$
with a, b, c,rational. Now it is pretty obvious that if you have two rational points, the line through them is a rational line. And it is neither hard to guess nor hard to prove that if you have two rational lines, then the point where they intersect is a rational point. If you have two linear equations with rational numbers as coefficients and you solve them, you get rational numbers as answers.

Keywords

Rational Number Elliptic Curve Rational Point Tangent Line Rational Line
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.