There are many good reasons to teach a course on a systematic introduction to symbolic methods not only to students of mathematics but also to those of technical sciences. The design of such a course meets an essential difficulty since the principles to be demonstrated appear only in non trivial applications in a convincing way, but there is no time to teach the necessary contexts to a large extend. Hence the material intended to demonstrate different effects has to be chosen with great care. The goal of this paper is to show that for such a course algebraic numbers are not only interesting by their mathematical content but also as a complex target where different concepts and principles of symbolic computations become apparent. Thus they may serve at once as a non trivial application of the basic concepts, notations and principles developed earlier in such a course.