Transcendentalism of curves and of numbers

Abstract

Lindemann's proof that the number e can not be the root of an algebraic equation with algebraic coefficients and algebraic exponents furnishes us with an example of a curve, namely y-e[superscirpt x] which has only one algebraic point, that is, only one point each of those coordinates is algebraic. We will divide points on a curve into three classes with respect to the character of their coordinates is algebraic. We will divide points on a curve into three classes with respect to the character of their coordinates: (1) totally algebraic, in which the x and y coordinates are both algebraic numbers, (2) totally transcendental, in which the x and y coordinates are both transcendental, (3) mixed, in which one coordinate is algebraic and other trascendental. With reference to the so called transcedentalism of the exponental curve Klein says, "Dieser Verlauf der exponential Kurve its gewiss eine hochst merkwurdige Tatsache." Klein was evidentaly thinking of the enormous amount of algebraic points in the plane, it was his first thought, just as it is ours, that a curve would have diffculty in avoiding them. The transcendentalism Klein spoke of is by no means "merkwurdig". There are uncountable hosts of curves with like pecularities. Let us consider first the algebraic ones. Before we begin our discussion, however, let us state a few facts about transcdental and algebraic numbers which will make what we shall say clearer.