**P. Philippon**

University of Paris, France

March 7, 2011

**Approximation of algebraic numbers by algebraic numbers of given degree**:
It is well known that irrational algebraic numbers are badly
approximated by rational numbers and this is best quantified in the celebrated Roth theorem. This theorem was extended to simultaneous approximation of several algebraic numbers as a consequence of Schmidt subspace theorem. Schmidt also extended Roth theorem to the approximation of an algebraic number by algebraic numbers of given smaller degree (over the field of rational numbers). We will present a result that combines both extensions, obtained in collaboration with Hans Peter Schlickewei. An interesting feature is that our
estimate shows a distinct behaviour in comparison with the subspace theorem, however it is not optimal with respect to the number of algebraic numbers considered.