منابع مشابه
Cohen–lenstra Heuristic and Roots of Unity
We report on computational results indicating that the well-known Cohen–Lenstra–Martinet heuristic for class groups of number fields may fail in many situations. In particular, the underlying assumption that the frequency of groups is governed essentially by the reciprocal of the order of their automorphism groups, does not seem to be valid in those cases. The phenomenon is related to the prese...
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This text was originally intended for non-mathematicians; during its preparation B. Mazur and A. Wiles announced the solution of the ’Main Conjecture’ in the theory of cyclotomic fields. It seemed appropriate to us to try to give an explanation of what this sensational discovery is all about.
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We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of roots of unity can be solved in polynomial time. As an auxiliary result, we solve the discrete logarithm problem for certain unit groups in finite...
متن کاملPrimitive Roots of Unity and Cyclotomic Polynomials1
We present a formalization of roots of unity, define cyclotomic polynomi-als and demonstrate the relationship between cyclotomic polynomials and unital polynomials. The following proposition is true (1) For every natural number n holds n = 0 or n = 1 or n ≥ 2. The scheme Comp Ind NE concerns a unary predicate P , and states that: For every non empty natural number k holds P [k] provided the fol...
متن کاملStable Computation of the Complex Roots of Unity Stable Computation of the Complex Roots of Unity
In this paper, we show that the problem of computing the complex roots of unity is not as simple as it seems at rst. In particular, the formulas given in a standard programmer's reference book Knuth, Seminumerical Algorithms, 1981] are shown to be numerically unstable, giving unacceptably large error for moderate sized sequences. We give alternative formulas, which we show to be superior both b...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2008
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2008.01.002