Algebraic and Transcendental Numbers

Abstract

We start this chapter by inverting the viewpoint of Chap. 15 . More precisely, we fix a complex number z and examine the set of polynomials $f\in \mathbb C[X]$ for which f(z) = 0. As a byproduct of our discussion, we give a (hopefully) more natural proof of the closedness, with respect to the usual arithmetic operations, of the set of complex numbers which are roots of nonzero polynomials of rational coefficients. We then proceed to investigate the special case of roots of unity, which leads us to the study of cyclotomic polynomials and allows us to give a partial proof of a famous theorem of Dirichlet on the infinitude of primes on certain arithmetic progressions. The chapter closes with a few remarks on the set of real numbers which are not roots of nonzero polynomials with rational coefficients.