A Survey of Diophantine Approximationin Fields of Power Series

Abstract.

 The fields of power series (or perhaps better called formal numbers) are analogues of the field of real numbers. Many questions in number theory which have been studied in the setting of the real numbers can be transposed to the setting of the power series. The study of rational approximation to algebraic real numbers has been intensively developped starting from the middle of the nineteenth century with the work of Liouville up to the celebrated theorem of Roth established in 1955. In the last thirty years, several mathematicians have studied diophantine approximation in fields of power series. We present here a summary of the present knowledge on this subject, emphasizing the analogies and differences with the situation in the real numbers case.