منابع مشابه
ROOTS OF UNITY AND NULLITY MODULO n
For a fixed positive integer , we consider the function of n that counts the number of elements of order in Zn. We show that the average growth rate of this function is C (logn) d( )−1 for an explicitly given constant C , where d( ) is the number of divisors of . From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo n of order is (d( )− 1)C (l...
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The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster. In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic fiel...
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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 کاملCohen–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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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2010
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-10-10341-4