By Henri Cohen

The computation of invariants of algebraic quantity fields equivalent to indispensable bases, discriminants, best decompositions, perfect category teams, and unit teams is necessary either for its personal sake and for its a number of functions, for instance, to the answer of Diophantine equations. the sensible com pletion of this activity (sometimes often called the Dedekind software) has been one of many significant achievements of computational quantity concept some time past ten years, because of the efforts of many folks. although a few useful difficulties nonetheless exist, you may ponder the topic as solved in a passable demeanour, and it truly is now regimen to invite a really good desktop Algebra Sys tem corresponding to Kant/Kash, liDIA, Magma, or Pari/GP, to accomplish quantity box computations that will were unfeasible purely ten years in the past. The (very quite a few) algorithms used are basically all defined in A direction in Com putational Algebraic quantity idea, GTM 138, first released in 1993 (third corrected printing 1996), that's observed right here as [CohO]. That textual content additionally treats different matters corresponding to elliptic curves, factoring, and primality trying out. Itis very important and normal to generalize those algorithms. numerous gener alizations will be thought of, however the most vital are definitely the gen eralizations to worldwide functionality fields (finite extensions of the sector of rational features in a single variable overa finite box) and to relative extensions ofnum ber fields. As in [CohO], within the current publication we are going to reflect on quantity fields purely and never deal in any respect with functionality fields.

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If a ( resp. , 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.

3. 3. Let x and y be two elements of an R-module M, and set v Then ax + by = abD - 1 x' + Dy' . 1 . Fundamental Results and Algorithms in Dedekind Domains 20 Proof. 4 with c = abil - 1 . 6. Let a, elements of K such that a - 1 il, this is clearly a special case of D b be two ideals. Assume that a, b, c, and d are jour ad - be = 1, a E a, b E b, c E b- 1 , d E a- 1 Let x and y be two elements of an R-module M, and set y' ) = (x y ) (x' Then ax (� �) + by = Rx ' + aby' . Proof. 4 with c = R and il = ab.

30. 24, the ideal class of D 1 · · · D n b 1 · · · b n is well-defined, hence also that of b 1 · · b n since the Di are unique. Finally, the ideal class of b 1 · · bm is well-defined, hence also that of b n H · · · b m. 29. 19 applied to the torsion-free module M'. Hence, we now assume that m = n, so MIN is a finitely generated torsion module. We prove the result by induction on n. Assume that n ;::: 1 and that it is true for n - 1 . 2. 19, we see that if b 1 = {x E Kl xw1 E M}, then M = b 1 w1 EB g(Mib 1 wi), where g is a section of the canonical projection of M onto Mlb 1 w1 .