Lattices and factorization of polynomials

• Authors: A. K. Lenstra
• Venue: ACM SIGSAM Bulletin
• Year: 1981

A new algorithm to factorize univariate polynomials over an algebraic number field has been implemented in Algol-68 on a CDC-Cyber 170-750 computer. The algebraic number field is given as the field of rational numbers adjoined by a root of a prescribed minimal polynomial. Unlike other algorithms [1,2] the efficiency of our so-called lattice algorithm does not depend on the irreducibility of the minimal polynomial modulo some prime. The factorization of the polynomial to be factored is constructed from the factorization of that polynomial over a finite field determined by a prime p and an irreducible factor of the minimal polynomial modulo p. The algorithm is based on a theorem on integral lattices and a theorem giving a lower bound for the length of a shortest-length polynomial having modulo pk a non-trivial common divisor with the minimal polynomial. These theorems also enable us to formulate a new algorithm for factoring polynomials over the integers. A technical report describing the algorithms will soon be available from the Mathematisch Centrum, Amsterdam.