Abstract. In this report we give an introduction to a constructive way of treating Systems of polynomial equations and inequalities. We present a method called cylindrical algebraic decomposition (CAD) discovered 1973 by Collins. The method constructs a decomposition of R" such that a given set of polynomials have constant sign on each component. All concepts needed to understand the algorithm is presented, e.g., polynomial remainder
sequences, subresultants, principal subresultant coefficients, Sturm chains and algebraic number representations.