Glimpses of Algebraic Geometry

Abstract

where F(x, y) is a polynomial with rational coefficients that is irreducible over the field Q of rational numbers. The set of points of the real plane 1R2 whose coordinates satisfy the equattion (1) is called a plane (rational) algebraic curve. If F is linear then we speak of a rational line. The points with rational coordinates are called rationalpoints. By the order of the curve r defined by equation (1) we mean the degree n of the polynomial F(x, y). The number of points of intersection of r and an arbitrary line AJc + By + C = O is exactly n. When counting the number of points of intersection we must consider multiplicities, complex points, and points at infinity. We give a few illustrative examples.