On Validated Computing in Algebraic Number Fields

In this paper we discuss several computations with elements of algebraic number fields F which either require numerical calculations or are at least speeded up considerably by their use. Since the results can be uniquely represented by (sequences of) rational integers, those numerical calculations should clearly be performed by validated methods. Hence, the intention of this paper is to point out important number theoretical problems which can be adequately solved only by the use of symbolic and validated computations.