Factorization of Polynomials Over Finite Fields

This paper discusses basic concepts in finite fields and emphasizes the meaning of irreducible polynomials and their relevance in algebraic analysis. The main focus is directed at the algorithms used to factor polynomials in finite fields through three important stages: distinct degree factorization, square-free factorization, and equal degree factorization. These stages are considered core procedures in determining the structure of polynomials and their relationship to more complex algebraic properties. Furthermore, this paper reviews the role of other algorithms that support this process, such as the Berlekamp algorithm, the Cantor–Zassenhaus algorithm, and several normalization techniques that enhance the effectiveness of the analysis. The combination of these various approaches allows the breakdown of polynomials into simpler factors, while also highlighting how the algorithms work synergistically to achieve accurate analysis results. Thus, this paper emphasizes the importance of a thorough understanding of polynomial factorization algorithms in finite fields, both in theory and application, and their contribution to the development of applied mathematics, particularly in the field of computational algebra.