Topics in Comptuter Algebra: Lecture 5
Computing with algebraic numbers.
Michael Monagan, Department of Mathematics, Simon Fraser University.
Abstract: We first show how to compute with algebraic numbers by computing modulo the minimal polynomial. Next we present some facts about cyclotomic polynomials and cyclotomic fields. Then we consider two algorithms for solving linear systems over cyclotomic fields. The first algorithm is ordinary Gaussian elimination. The second algorithm is a modular algorithm that uses Chinese remaindering and rational number reconstruction. We also use primes which split the minimal polynomial so that we can reduce the computation to solving linear systems over the integers modulo a machine prime.