By Henri Cohen

The computation of invariants of algebraic quantity fields comparable to imperative bases, discriminants, major decompositions, excellent classification teams, and unit teams is critical either for its personal sake and for its a variety of functions, for instance, to the answer of Diophantine equations. the sensible com pletion of this job (sometimes often called the Dedekind application) has been one of many significant achievements of computational quantity idea some time past ten years, because of the efforts of many folks. although a few functional difficulties nonetheless exist, you will contemplate the topic as solved in a passable demeanour, and it's now regimen to invite a really expert desktop Algebra Sys tem akin to 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 basically all defined in A direction in Com putational Algebraic quantity idea, GTM 138, first released in 1993 (third corrected printing 1996), that is pointed out right here as [CohO]. That textual content additionally treats different topics corresponding to elliptic curves, factoring, and primality checking out. Itis vital and common to generalize those algorithms. numerous gener alizations may be thought of, however the most crucial are definitely the gen eralizations to international functionality fields (finite extensions of the sector of rational services in a single variable overa finite box) and to relative extensions ofnum ber fields. As in [CohO], within the current booklet we'll contemplate quantity fields simply and never deal in any respect with functionality fields.

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6. (6) Computing the intersection M n N of two modules is slightly more dif ficult. 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) .

We also have the following simple proposition. 34. Assume that there exist nonzero ideals ai such that an R-module M satisfies M � ffi 1 < i < k R/ai . Then the order-ideal of M is equal -to rr l� i ::;k ai . 16 1 . Fundamental Results and Algorithms in Dedekind Domains Proof. This immediately follows from the fact that the order-ideal is un changed by module isomorphism, and that the order-ideal of a product of two modules is equal to the product of the order-ideals. D We end this section with the elementary divisor theorem for torsion-free modules, which is now easy to prove using the above techniques.

1 0 . Let a be an integral ideal of R and a E a, a -:j; 0. Assume that the prime ideal factorization of a is known. Then there exists a polynomial-time algorithm that finds b E a such that a = aR + bR. Proof. Write a R = TI P p e p with e p 2: 0. Thus, a = TI P pv p ( a ) with 0 ::; vp (a) ::; e p . 8 we can, in polynomial time, find b E R such that vp (b) = vp (a) for all p I a; by looking at p-adic valuations, it is clear that D a = aR + bR. 22 1. Fundamental Results and Algorithms in Dedekind Domains Remarks Recall that R is the ring of integers of a number field.