The role of complex conjugation in transcendental number theory

In his two well known 1968 papers “Contributions to the theory of transcendental numbers”, K. Ramachandra proved several results showing that, in certain explicit sets {x1, . . . , xn} of complex numbers, one element at least is transcendental. In specific cases the number n of elements in the set was 2 and the two numbers x1, x2 were both real. He then noticed that the conclusion is equivalent to saying that the complex number x1 + ix2 is transcendental. In his 2004 paper published in the Journal de Théorie des Nombres de Bordeaux, G. Diaz investigates how complex conjugation can be used for the transcendence study of the values of the exponential function. For instance, if log α1 and logα2 are two non-zero logarithms of algebraic numbers, one of them being either real of purely imaginary, and not the other, then the product (logα1)(logα2) is transcendental. We will survey Diaz’s results and produce further similar ones. 1 Theorems of Hermite–Lindemann and Gel’fond–Schneider Denote by Q the field of algebraic numbers and by L the Q-vector space of logarithms of algebraic numbers: L = {λ ∈ C ; e ∈ Q} = exp−1(Q) = {logα ; α ∈ Q}. The Theorem of Hermite and Lindemann ([13], Theorem 1.2) can be stated in several equivalent ways: ∗Many thanks to N. Saradha for the perfect organization of this Conference DION2005 and for taking care in such an efficient way of the publication of the proceedings.