Algorithms for solving linear systems over cyclotomic fields

We consider the problem of solving a linear system Ax = b over a cyclotomic field. What makes cyclotomic fields of special interest is that we can easily find a prime p that splits the minimal polynomial m(z) for the field into linear factors. This makes it possible to develop very fast modular algorithms. We give two output sensitive modular algorithms, one using multiple primes and Chinese remaindering, and the other using linear p−adic lifting. Both of our algorithms use rational reconstruction to recover the rational coefficients in the solution vector. We have implemented both algorithms in Maple with key parts of the implementation implemented in C for efficiency. A complexity analysis and experimental timings both show that Chinese remaindering is competitive with p−adic lifting. We also give a third algorithm which computes the solution x as a ratio of two determinants modulo m(z) using Chinese remaindering. This representation is a factor of d = degm more compact in general, and because of this, we can compute it the fastest in general.