By Henri Cohen
The computation of invariants of algebraic quantity fields equivalent to indispensable bases, discriminants, leading decompositions, excellent category teams, and unit teams is critical 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 job (sometimes often called the Dedekind software) has been one of many significant achievements of computational quantity idea some time past ten years, due to the efforts of many folks. even supposing a few sensible difficulties nonetheless exist, you will think of the topic as solved in a passable demeanour, and it's now regimen to invite a really good machine Algebra Sys tem reminiscent of Kant/Kash, liDIA, Magma, or Pari/GP, to accomplish quantity box computations that may were unfeasible simply ten years in the past. The (very various) algorithms used are primarily all defined in A path in Com putational Algebraic quantity concept, GTM 138, first released in 1993 (third corrected printing 1996), that is spoke of right here as [CohO]. That textual content additionally treats different matters comparable to elliptic curves, factoring, and primality trying out. Itis vital and traditional to generalize those algorithms. numerous gener alizations could 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 ebook we are going to give some thought to quantity fields simply and never deal in any respect with functionality fields.
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Additional info for Advanced Topics in Computational Number Theory
Following the classical proof (see, for example, [CohO, Proposition 4. 3. 7, we can in polynomial time find a p E ap whose sum is equal to 1 . Furthermore, we can find bp E p e p ' p e p + l (for example, by taking the epth power of an element of p , p 2 which can be found in polynomial time) . Then a = L pES a p b p is a D solution to our problem. 9. Given two integral ideals a and b of R such that the fac torization of the norm of b is known, there exists a polynomial-time algorithm that finds x E K such that xa is an integral ideal coprime to b, and similarly finds y E K such that y a 1 is an integral ideal coprime to b .
Depending on the context, however, there may be faster methods. ( 4) To compute the product M N when this makes sense, we form all the possible products of the generators and their corresponding ideals, and compute the HNF of the resulting pseudo-matrix. Usually, however, there are faster methods. 6) and if N is given in HNF, we must only multiply the generators and ideals of N by the two pseudo elements of M . ( 5 ) To compute the image and the kernel of a map f from N to M, we proceed as follows.
There exist E a() - 1 and E bll - 1 such that au + bv = 1 , and these elements can be found in polynomial time. u v u, v Proof. 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.
Advanced Topics in Computational Number Theory by Henri Cohen