Galois theory, motives and transcendental numbers

Abstract

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, could one not associate conjugates and a Galois group to transcendental numbers such as π? Beyond, can one not 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.