代数扩张域上多项式因式分解的一个有效算法
An Efficient Algorithm for Factoring Polynomials over Algebraic Extension Field
摘要：
提出了代数扩张域上多项式因式分解的一个新的有效算法。扩张域由模极大理想的多项式环定义。如果极大理想由它的Groebner基给出，则不需要额外的Groebner基计算来分解这个扩张域上的多项式。利用线性代数技巧，通过一般的线性映射，得到基场上的多项式。然后这个多项式在基场上被因式分解。利用这些因子，得到了扩张域上多项式的因子分解。新算法已被实现，计算机实验表明，新算法是非常有效的，特别是在复杂的例子。
英文摘要：
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra Groebner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a polynomial over the ground field by a generic linear map. Then this polynomial is factorized over the ground field. From these factors, the factorization of the polynomial over the extension field is obtained. The new algorithm has been implemented and computer experiments indicate that the new algorithm is very efficient, particularly in complicated examples.
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