Polynomials over Z_n

Más información sobre el libro

Polynomials are a fundamental topic in mathematics, originating from the study of real and complex functions. Recent research has expanded the exploration of polynomials beyond traditional realms, including commutative rings with identity and various algebraic structures like groups and lattices. This work provides a comprehensive overview of polynomials in commutative rings with identity, focusing particularly on the ring Z_n, or residue classes modulo n. It covers essential definitions and established results while introducing readers to the abstract theory of polynomials, which aids in understanding the solvability of polynomial equations of the form f = 0 over a variety V. The text further explores when such equations have solutions in Z_n or its extension rings, alongside discussions on reducible and irreducible polynomials within Z_n. Later chapters delve into Galois theory for finite local commutative rings, identifying conditions under which the factor ring Z_n/(f) serves as the smallest extension of Z_n. Additionally, readers will learn about representing functions from Z_n to Z_n as polynomial functions. This work is aimed at students and anyone interested in grasping key aspects of polynomial theory, particularly in relation to Z_n.