By Henri Cohen

The computation of invariants of algebraic quantity fields resembling imperative bases, discriminants, leading decompositions, perfect classification teams, and unit teams is necessary either for its personal sake and for its quite a few 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 conception long ago ten years, due to the efforts of many folks. even supposing a few sensible difficulties nonetheless exist, possible reflect on the topic as solved in a passable demeanour, and it really is now regimen to invite a really good machine Algebra Sys tem similar to Kant/Kash, liDIA, Magma, or Pari/GP, to accomplish quantity box computations that might were unfeasible basically ten years in the past. The (very quite a few) algorithms used are basically all defined in A direction in Com putational Algebraic quantity thought, GTM 138, first released in 1993 (third corrected printing 1996), that is talked about right here as [CohO]. That textual content additionally treats different topics equivalent to elliptic curves, factoring, and primality trying out. Itis vital and normal to generalize those algorithms. a number of gener alizations may be thought of, however the most vital are definitely the gen eralizations to international functionality fields (finite extensions of the sphere of rational services in a single variable overa finite box) and to relative extensions ofnum ber fields. As in [CohO], within the current e-book we are going to think about quantity fields basically and never deal in any respect with functionality fields.

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3. 11), we can find Yi E 'IlK such that vp (ad O:i -Yi) � 1 . Hence in M/ pM we have x = L i Yi"li, which is the desired decomposition. 22, see Exercise 13) . 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.

Thus, if Vj is the jth column of UI , we have (c1Vj)i c ai, hence c1AVj c M since (A, (ai)) is a pseudo-matrix for M. It follows that the module defined by (HI , JI ) is a submodule of M. But since BU3 = -AUI , the same reasoning on B and U3 shows that it is also a submodule of N; hence it is a submodule of M n N. Conversely, an element of M n N can be represented as AX = -BY for some vectors X = (xi) and Y = (yj) such that Xi E ai and Y1 E bj . 6 (5) tells us that this vector will be in the image of ( � ) , JI ) .

Let a be the annihilator of M in R, so that a = {x E Rl xM = {0} } . Clearly, a is an R-module contained in R, hence is an integral ideal, and it is nonzero since M is a finitely generated torsion module (it is the intersection of the annihilators of some generators of M, hence a finite intersection of nonzero ideals) . 32 above. Then B is a principal ideal domain. Furthermore, if x E B, then x = n I d with ( dR, a) = 1 ; hence dR + a = R. Multiplying by M, we obtain dM = M, hence M = MId, and so xM = nMid C M; hence BM C M, and so BM = M since 1 E B.