We describe in detail the implementation of an algorithm which computes the class group and the unit group of a general number field, and solves the principal ideal problem. The basic ideas of this algorithm are due to J. Buchmann. New ideas are the use of LLL-reduction of an ideal in a given direction which replaces the notion of neighbour, and the use of complex logarithmic embeddings of elements which plays a crucial role. Heuristically the algorithm performs in sub-exponential time with respect to the discriminant for fixed degree, and performs well in practice.
Following the terminology of Wallace [8] we shall use the word mob to mean a Hausdorff topological semigroup, and shall use clan for a compact connected mob with unit. Interval means a closed interval on the real line, although as A. H. Clifford has pointed out to the authors, nearly all the theorems (and proofs) generalize to arbitrary compact connected linearly ordered topological spaces.The object of this paper is to characterize clans with zero on an interval. Partial results in this connection have been found by Faucett [3; 4] and Clifford [l]. In addition the case when 0 (the zero) is an end point has been studied by Mostert and Shields [5]. Finally a forthcoming paper of Clifford In what follows 5 is always a clan on an interval with zero. It is well known (e.g. Wallace [7]) that the unit u is an end point. We will assume that it is the right hand end point (the other case, of course, can be handled by a dual argument) and call the other end point 5. Let L be the interval [S, 0] and R the interval [0, u] so that we have the following diagram for S: L R S 0 u Fig. 1 We define a partial order ■< on 5 as follows: x
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