By Henri Cohen
The computation of invariants of algebraic quantity fields corresponding to essential bases, discriminants, best decompositions, excellent category teams, and unit teams is necessary either for its personal sake and for its various purposes, for instance, to the answer of Diophantine equations. the sensible com pletion of this job (sometimes referred to as the Dedekind software) has been one of many significant achievements of computational quantity conception some time past ten years, due to the efforts of many of us. even if a few functional difficulties nonetheless exist, possible reflect on the topic as solved in a passable demeanour, and it truly is now regimen to invite a really expert machine Algebra Sys tem similar to Kant/Kash, liDIA, Magma, or Pari/GP, to accomplish quantity box computations that might were unfeasible in basic terms ten years in the past. The (very various) algorithms used are basically all defined in A direction in Com putational Algebraic quantity conception, GTM 138, first released in 1993 (third corrected printing 1996), that's noted right here as [CohO]. That textual content additionally treats different topics reminiscent of elliptic curves, factoring, and primality trying out. Itis very important and ordinary to generalize those algorithms. numerous gener alizations should be thought of, however the most vital are definitely the gen eralizations to international functionality fields (finite extensions of the sector of rational capabilities in a single variable overa finite box) and to relative extensions ofnum ber fields. As in [CohO], within the current e-book we'll give some thought to quantity fields purely and never deal in any respect with functionality fields.
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Additional info for Advanced Topics in Computational Number Theory
5 . p of a Pseudo-Basis Another natural question is the following. Assume that M is a finitely gener ated, torsion-free module, and that, thanks to the above algorithms, we have written M = E9 i aiWi in terms of a pseudo-basis. Let p be a prime ideal of R. Then MjpM is in a natural manner a vector space over the field k = Rjp , and we can ask for a basis of this vector space over k. This can be done using the following algorithm. (wi, ai) be a pseudo-basis for a fi nitely generated , torsion-free R-module M, and let p be a (nonzero) pri me idea l of R.
32 1 . Fundamental Results and Algorithms in Dedekind Domains Proof. Trivial and left to the reader. D Corollary 1 . 4. 1 1 . For each i and j with 1 s i < j s n, let Si,i be a system of representatives of K/cicj 1 . 6. Then in that theorem, we may assume that for every i and j such that i < j the entry in row i and column j of the matrix H is in Si,j , in which case the matrix H is unique. Proof. For i < j, let hi,i be the entry in row i and column j of the matrix H. There exists a unique h�. i E Si,j such that q = hii,j - h i,j E CiCj- l .
In [CohO, Exercise 18 of Chapter 4] , we have given a possible so lution. However, the following algorithm is more elegant and useful also over z . (I thank D. ) Algorithm 1 . 5 . 1 (Intersection of Modules) . Let M and N be two modu les of the sa me ra n k n given by some pseudo-generati ng sets. This algorithm computes a n H N F pseudo-basis for M n N. 1 . 7, compute the H N F (A, I) and (B, J) of the modu les M and N, with I = (ai) and J = (b1) ( only a pseudo-basis i s necessary, not the H N F) .
Advanced Topics in Computational Number Theory by Henri Cohen