Polynomials over Z_n

ISBN-13: 9783836424332 / Angielski / Miękka / 2007 / 72 str.

Jorg Forstner

Opis

Polynomials are a classical subject of mathematics. The first step towards theabstract concept of polynomials was the investigation of real and complexfunctions.During the last century studies on polynomials over commutativerings with identity gave rise for researches on polynomials over other classesof algebraic structures, such as groups, semigroups,lattices, rings (with orwithout identity), etc.. The purpose of this work is to give an overview ofpolynomials over commutative rings with identity and especially over the ringZ_n, the residue classes modulo n. Besides basic definitions and well-knownresults,the reader will be introduced to aspects of the abstract theory ofpolynomials to have results for the solvability of a polynomial equation of theform f = 0 over a variety V. This will be the basis to establish results when apolynomial equation f = 0 over the ring Z_n has got a solution in Z_n or in anextension ring of Z_n. Further more, the reader will find results on reducibleand irreducible polynomials over Z_n. The last chapters deal with the Galoistheory for finite local commutative rings and we will see when the factor ringZ_n[x]/(f) is the smallest extension ring of Z_n. Eventually, the reader will getto know some results when a function f from Z_n to Z_n can be representedas a polynomial function over Z_n. The book addresses to students who areinterested in the subject as well as all interested people who want to getfamiliar with the most important aspects of the theory of polynomials andespecially with polynomials over Z_n.