Topics in Comptuter Algebra: Lecture 5

Computing with algebraic numbers.

Michael Monagan, Department of Mathematics, Simon Fraser University.

Wednesday July 18, 2007, K9509, 2:30-4:30pm.

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.