"Symbolic computations with algebraic numbers" Hans-Gert Graebe Universitaet Leipzig, Institut f. Informatik Augustusplatz 10/11 D-04109 Leipzig Germany E-mail:graebe@informatik.uni-leipzig.de Abstract: The aim of the talk is to show the role of algebraic numbers in symbolic computations, advantages of the RootOf notation, trigonometric algebraic numbers, some mathematics behind that, algebraic extension towers, and what and how these things (currently) may be done with the different general purpose CAS (i.e. Axiom, Derive, Macsyma, Maple, Mma, MuPAD, Reduce).