Details:
Title  Algorithms for solving linear systems over cyclotomic fields  Author(s)  Liang Chen, Michael Monagan  Type  Article in Journal  Abstract  We consider the problem of solving a linear system A x = b over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime p for which the minimal polynomial m ( z ) for the field factors into a product of distinct linear factors. This makes it possible to develop fast modular algorithms. We give two output sensitive modular algorithms, one using multiple primes and Chinese remaindering, the other using linear p adic lifting. Both use rational reconstruction to recover the rational coefficients in the solution vector. We also give a third algorithm which computes the solutions as ratios of two determinants modulo m ( z ) using Chinese remaindering only. Because this representation is d = deg m ( z ) times more compact in general, we can compute it the fastest. We have implemented the algorithms in Maple. Our benchmarks show that the third method is fastest on random inputs, but on real inputs arising from problems in computational group theory, the first two methods are faster because the solutions have small rational coefficients.  Keywords  Linear systems, Modular algorithms, Cyclotomic fields, Cyclotomic polynomials  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717110000714 
Language  English  Journal  Journal of Symbolic Computation  Volume  45  Number  9  Pages  902  917  Year  2010  Edition  0  Translation 
No  Refereed 
No 
