Factoring multivariate polynomials over algebraic number fields

Abstract

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented in the algebraic manipulation system MACSYMA. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗<!-- ∗ --></mml:mo> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">^{\ast \ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Some machine examples with timing are included.