By Henri Cohen
The computation of invariants of algebraic quantity fields equivalent to necessary bases, discriminants, leading decompositions, perfect classification 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 activity (sometimes often called the Dedekind software) has been one of many significant achievements of computational quantity concept long ago ten years, due to the efforts of many of us. even supposing a few sensible difficulties nonetheless exist, you can actually think of the topic as solved in a passable demeanour, and it really is now regimen to invite a really expert computing device Algebra Sys tem resembling Kant/Kash, liDIA, Magma, or Pari/GP, to accomplish quantity box computations that may were unfeasible in basic terms ten years in the past. The (very a number of) algorithms used are primarily all defined in A direction in Com putational Algebraic quantity thought, GTM 138, first released in 1993 (third corrected printing 1996), that is mentioned right here as [CohO]. That textual content additionally treats different topics equivalent to elliptic curves, factoring, and primality checking out. Itis vital and traditional to generalize those algorithms. a number of 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 services in a single variable overa finite box) and to relative extensions ofnum ber fields. As in [CohO], within the current ebook we'll think about quantity fields simply and never deal in any respect with functionality fields.
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Before doing this, however, let us see how one can go from one basis to another. 4. 2. Let (wi, ai)i and (TJj , bj)j be two pseudo-bases for an R module M, and let U = (ui,j) be the n x n matrix giving the 1Ji in terms of the Wi (so that (TJI , . . , 1Jn ) = (w1 , . . , Wn ) U) . Set a = a1 · · · an and b = b 1 · · · b n . Then ui,j E ai b j 1 and a = det(U) b (note that, by Theorem 1 . 2. 25, we know that a and b are in the same ideal class) . Conversely, if there exist ideals b i such that a = det(U) b (with b = b 1 · · · b n ) and Ui,j E ai bj 1 , then (TJj , bj)j is a pseudo-basis of M, where the 1Ji are given in terms of the Wi by the columns of U.
In the absolute case where M = ZK is the ring of integers of a number field K considered as a Z-module and T is•the trace, the discriminant ideal DT(M) gives the absolute value of the usual discriminant, and dT (M) gives its sign (and some other information already contained in DT (M)) . Since we represent finitely generated, torsion-free modules by pseudo bases, we must also explain how to represent linear maps between such mod ules. 2. 4. Let (wi , ai)i be a pseudo-basis for a finitely generated, torsion-free module M, and similarly (wj , aj )j for a module M' .
Note that in practice, n will be the relative degree of number fields extensions, and so in many cases the naive algorithm will be sufficient. 6. We first need a definition. 8. Let (A, I) be a pseudo-matrix with I = (aj ) · If i 1 , . . , i r are r distinct rows of A and ii , . . , ir are r distinct columns, we define the minor-ideal corresponding to these indices as follows. Let d be the determinant of the r x r minor extracted from the given rows and columns of A. Then the minor-ideal is the ideal daj1 • • • air .
Advanced Topics in Computational Number Theory by Henri Cohen