1) Prime number tests, factorization in Z
2) LLL-algorithums
3) Polynomial factorization over finite fields over Z, Q or in K [x1,...,xn]
4) Gröbner bases, resultatants, eliminations
5) Primary decompostion, radical ideals, Syzygies and free resolutions
6) Practical applications, such as the examination of processors, states of balance in economic models, the description of configuration spaces in molecules, robotics or Sudoku
For all topics the emphasis is on practical work using a concrete computer-algebra system (such as Singular).