Reduced Ideals from the Reduction Algorithm

The reduction algorithm is used to compute the reduced ideals of a number field. However, there are reduced ideals that can never be obtained from this algorithm. In this paper, we will show that these ideals have inverses of larger norms among reduced ones. Especially, for some number fields, we present the conditions of the reduced ideals produced by the reduction algorithm. Additionally, our results partly answer the question by R. Schoof who asks about the number of the reduced ideals that can not be produced by the reduction algorithm.