By Henri Cohen
The computation of invariants of algebraic quantity fields similar to quintessential bases, discriminants, leading decompositions, perfect category teams, and unit teams is necessary either for its personal sake and for its a variety of purposes, for instance, to the answer of Diophantine equations. the sensible com pletion of this activity (sometimes referred to as the Dedekind software) has been one of many significant achievements of computational quantity idea long ago ten years, because of the efforts of many folks. even if a few useful difficulties nonetheless exist, you possibly can ponder the topic as solved in a passable demeanour, and it really is now regimen to invite a really good machine Algebra Sys tem resembling Kant/Kash, liDIA, Magma, or Pari/GP, to accomplish quantity box computations that might were unfeasible purely ten years in the past. The (very a variety of) algorithms used are primarily all defined in A direction in Com putational Algebraic quantity conception, GTM 138, first released in 1993 (third corrected printing 1996), that is talked about the following as [CohO]. That textual content additionally treats different topics corresponding to elliptic curves, factoring, and primality checking out. Itis very important and normal to generalize those algorithms. numerous gener alizations should be thought of, however the most crucial are definitely the gen eralizations to international functionality fields (finite extensions of the sphere of rational features in a single variable overa finite box) and to relative extensions ofnum ber fields. As in [CohO], within the current booklet we'll think of 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
1 / a E a() - 1 = R/a) . S o assume a and b are nonzero. Set I = a a() - 1 and J = b b ll - 1 . By the definition of () - 1 , I and J are integral ideals and we have I + J = R. 1, we can thus find in polynomial time e E I and f E J such that e + f = 1 , and clearly = efa and = f / b satisfy the conditions of the lemma. 3 Basic Algorithms in Dedekind Domains 19 Remark. Although this proposition is very simple, we will see that the essential conditions u E aD - 1 and E bD - 1 bring as much rigidity into the problem as in the case of Euclidean domains, and this proposition will be regularly used instead of the extended Euclidean algorithm.
6. 1 Introduction It is well known that the usual HNF over 7l suffers from coefficient explosion, which often makes the algorithm quite impractical, even for matrices of rea sonable size. Since our algorithm is a direct generalizati;n of the naive HNF algorithm, the same phenomenon occurs. Hence, it is necessary to improve the basic algorithm. In the case of the ordinary HNF, there are essentially two ways of doing this, depending on what one wants. The first method is the "modular" method. 6] ) .
Fundamental Results and Algorithms in Dedekind Domains 14 Proof. Let x = nld with n and d in R, for the moment arbitrary. By the approximation theorem, there exists b E K such that 'v'p E S, Vp (b) = -vp (d) and 'v'p tj_ S, Vp (b) 2: 0 It follows that for p E S, vp (db) = 0 and for p tj. S, vp (db) 2: 0, so db E R and is not divisible by any p in S. Since for all p E S, vp (x) 2: 0 or, equivalently, vp (n) 2: vp (d) , it follows that vp (nb) 2: vp (db) = 0 for p E S and vp (nb) 2: 0 for p tj. S, hence nb E R, so x = (nb)l (db) is a suitable representation of 0 x.
Advanced Topics in Computational Number Theory by Henri Cohen