Polynomial Factorization

• Published in: Computer Algebra and Symbolic Computation pp. 367 - 448
• Main Author: Cohen, Joel S.
• Format: Book Chapter
• Language: English
• Published: United Kingdom A K Peters/CRC Press 2003; CRC Press LLC
• Subjects: Basis Polynomial; Multivariate Polynomials; Prime; Monic Polynomial; Polynomial Division; Chinese Remainder Theorem; Pivot Entry; Symmetric Representation; Auxiliary Polynomials; Irreducible Factorization; Gcd Calculations; Elimination Step; Irreducible Polynomial; Lifting Step; Positive Degree; Extended Euclidean Algorithm; Computer Algebra Systems; Square Free Polynomial; Primitive Polynomial; Polynomial Factorization; Trial Factors; Linearly Independent; Multivariable Polynomials; Factorization Algorithm
• ISBN: 9781568811598; 1568811594
• DOI: 10.1201/9781439863701-15

The goal of this chapter is the complete description of a modern algorithm for the factorization of polynomials in Q[x] in terms of irreducible polynomials.In Section 9.1 we describe an algorithm that obtains a partial factorization of a polynomial. The algorithm can separate factors of different multiplicities as inbut is unable to separate factors of the same multiplicity as inThis factorization is important, however, because it reduces the factorization problem to polynomials without multiple factors. In Section 9.2 we describe the classical approach to factorization, which is known as Kronecker's algorithm. This algorithm is primarily of historical interest because it is much too slow to be used in practice. In Section 9.3 we describe an algorithm that factors polynomials in Zp[x). Although this algorithm is important in its own right, it is included here because it plays a role in the modern approach for factorization in Q[x}. Finally, in Section 9.4 we describe a modern factorization algorithm, known as the Berlekamp-Hensel algorithm, which uses a related factorization in Zp[x] together with a lifting algorithm to obtain the factorization in Q[x\.