• 论文 •

### 连续代数扩域上多项式因式分解的Trager算法

1. 中科院数学与系统科学研究院数学机械化重点实验室, 北京 100080
• 收稿日期:1900-01-01 修回日期:1900-01-01 出版日期:2006-10-25 发布日期:2006-10-25

Yuan Chunming. Trager's Factorization Algorithm over Successive Extension Fields[J]. Journal of Systems Science and Mathematical Sciences, 2006, 26(5): 533-540.

### Trager's Factorization Algorithm over Successive Extension Fields

Yuan Chunming

1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080
• Received:1900-01-01 Revised:1900-01-01 Online:2006-10-25 Published:2006-10-25

Polynomial factorizations are basic problems in symbolic computation. Polynomial factorization algorithms appeared in the 1960's are considered to be the origin of the field of symbolic computation. At present, polynomial factorization algorithms are well established and implemented in symbolic computation software such as MAPLE. But factorization algorithms over successive algebraic extension fields are still under investigation. The basic factorization algorithm over algebraic extension fields is Trager's algorithm. Algorithms for a single algebraic extension field based
on Hensel lifting are given by Weinberger et al. However, in order to compute the irreducible ascending chain in Wu's method, polynomial factorizations over successive algebraic extension fields are needed. Wu, Hu, and Wang independently put forward factorization algorithms over successive algebraic extension fields based on methods of equation solving. Similar to the Trager's algorithm, Wang and Lin proposed another algorithm reducing the problem to the
factorization over the rational number field. In their approach, Wu's triangularization algorithm is used, and hence the termination of the algorithm depends on the computation of Wu's method. Zhi applied the lifting technique to the factorization over successive algebraic extension fields. A direct algorithm on factorization over successive algebraic extension fields is given in this paper,
extending Trager's algorithm to factorization over successive algebraic extension fields. The proposed algorithm only uses resultant computation and factorization over the rational number field.

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