Jörg Forstner

Polynomials over Z_n Jörg Forstner

Polynomials are a classical subject of mathematics. The first step towards the abstract concept of polynomials was the investigation of real and complex functions. During the last century studies on polynomials over commutative rings with identity gave rise for researches on polynomials over other classes of algebraic structures, such as groups, semigroups,lattices, rings (with or without identity), etc.. The purpose of this work is to give an overview of polynomials over commutative rings with identity and especially over the ring Z_n, the residue classes modulo n. Besides basic definitions and well-known results,the reader will be introduced to aspects of the abstract theory of polynomials to have results for the solvability of a polynomial equation of the form f = 0 over a variety V. This will be the basis to establish results when a polynomial equation f = 0 over the ring Z_n has got a solution in Z_n or in an extension ring of Z_n. Further more, the reader will find results on reducible and irreducible polynomials over Z_n. The last chapters deal with the Galois theory for finite local commutative rings and we will see when the factor ring Z_n[x]/(f) is the smallest extension ring of Z_n. Eventually, the reader will get to know some results when a function f from Z_n to Z_n can be represented as a polynomial function over Z_n. The book addresses to students who are interested in the subject as well as all interested people who want to get familiar with the most important aspects of the theory of polynomials and especially with polynomials over Z_n.