Book Review: Transcendental numbers

Abstract

1The title may suggest that the book deals with the general theory of transcendental numbers. A complex number a is said to be algebraic if it is a root of a polynomial f{x) = a n x n H h a x x + a 0 with rational coefficients and ƒ (JC) ^ 0. If a is not algebraic, it is called transcendental. In 1874, Cantor showed that the set of all algebraic numbers is countable so that transcendental numbers exist. The first rigorous proof of the existence of transcendental numbers was given thirty years earlier by Liouville. We say that a is of degree n, if the smallest degree of polynomials ƒ as described above equals n. Liouville proved the existence of a positive constant c(a) such that every pair of rational integers p, q with q > 0 and p/q ^ a satisfies