Advanced Topics in Computational Number Theory by Henri Cohen

By Henri Cohen

The computation of invariants of algebraic quantity fields comparable to imperative bases, discriminants, leading decompositions, perfect category teams, and unit teams is necessary either for its personal sake and for its a variety of purposes, for instance, to the answer of Diophantine equations. the sensible com­ pletion of this job (sometimes referred to as the Dedekind software) has been one of many significant achievements of computational quantity thought long ago ten years, due to the efforts of many folks. even supposing a few useful difficulties nonetheless exist, it is easy to contemplate the topic as solved in a passable demeanour, and it truly is now regimen to invite a really good machine Algebra Sys­ tem equivalent to Kant/Kash, liDIA, Magma, or Pari/GP, to accomplish quantity box computations that will were unfeasible merely ten years in the past. The (very a number of) 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's stated the following as [CohO]. That textual content additionally treats different matters similar to elliptic curves, factoring, and primality trying out. Itis vital and typical to generalize those algorithms. a number of 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 booklet we'll ponder quantity fields in basic terms and never deal in any respect with functionality fields.

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Given two fractional idea ls a and b, this a lgorithm outputs four elements a, b, e, and d such that a E a, b E b, e E b - 1 , d E a - 1 , and ad - be = 1 . 1 . [Remove denomi nators] Let d1 E Q (or even i n K) b e a common denominator for the generators of a, and similarly d2 for b, and set a +-- d1 a, b +-- d2 b. 2 . [LLL-red uce] Using a n LLL-a lgorithm, compute a n LLL-reduced basis (ai) of a. 3. 16 4. 5. 6. 7. some fixed i ntegra l basis. 13, pick a sma ll ra ndom element a E a. [Check if O K] Form the m x 2m matrix M whose first m columns give the product of a by the basis elements of a - 1 , and the last m columns give a Z-basis of b on the fixed i ntegra l basis.

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. Proof. Since n n 1 1 1Ji E bj M = b j EB aiwi = EB ai bj 1 wi , i= l i= l it follows that ui,i E ai b j 1 It is easily proven by linearity or by induction on n that e = det(U) E a b - 1 so e b c a.

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 . With this definition we can state the following result. 9. 4. 6, for 1 � j � n, set c1 = b k - n+i . Then the ideals Cj are unique. More precisely, if we call gi = g1 (A) the ideal generated by all the (n + 1 - j) x (n + 1 - j) minor-ideals in the last n + 1 - j rows of the matrix A, then Cj = g n+l - ig;;-�1 . Proof. 4. 7. In particular, g1 (A) = g1 (AU) .

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