Galois Theory, Motives and Transcendental Numbers

From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn’t one associate conjugates and a Galois group to transcendental numbers such as π? Beyond, can’t one envision an appropriate Galois theory in the field of transcendental number theory? In which role? The aim of this text is to indicate what Grothendieck’s theory of motives has to say, at least conjecturally, on these questions. 1. THE BASIC QUESTION. Let α be an algebraic complex number: this means that α is a root of a non-zero polynomial p with rational coefficients. One may assume that p is of minimal degree, say n; this ensures that p has no multiple roots. Its complex roots are called the conjugates of α. The polynomial expressions with rational coefficients in the conjugates of α form a field (the splitting field of p), also called the Galois closure of Q[α]. We denote it by Q[α]gal and view it as a subring of C. The Galois group of α (or p) is the group of automorphism of the ring Q[α]gal. We denote it by Gα. Two fundamental facts of Galois theory are: (1) Gα identifies with a subgroup of the permutation group of the conjugates of α, and permutes transitively these conjugates, (2) the elements in Q[α]gal fixed by Gα are in Q. In this paper, we address the following 1.0.1. Basic question. Is there anything analogous for (some) transcendental numbers? 1991 Mathematics Subject Classification. 32G, 14D, 11J, 34M.