Finite Commutative Rings and Their Applications

Finite Commutative Rings and Their Applications by Gilberto Bini, Flaminio Flamini.

Foreword by Dieter Jungnickel Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. This book will be of interest to both professional and academic researchers in the fields o...

Full description

Saved in:
Bibliographic Details
Main Authors: Bini, Gilberto (Author), Flamini, Flaminio (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY : Springer US : Imprint: Springer, 2002.
Edition:1st ed. 2002.
Series:The Springer International Series in Engineering and Computer Science, 680
Springer eBook Collection.
Subjects:
Electrical engineering.
Coding theory.
Information theory.
Computer science—Mathematics.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1 Fundamental Notions In Ring Theory
  • 1.1 Basic Definitions
  • 1.2 Prime and Maximal Ideals
  • 1.3 Euclidean Domains, P.I.D.’s and U.F.D.’s
  • 1.4 Factorization in $$ {Z_p}n $$[x]
  • 2 Finite Field Structure
  • 2.1 Basic Properties
  • 2.2 Characterization of Finite Fields
  • 2.3 Galois Field Automorphisms
  • 3 Finite Commutative Rings. Regular Polynomials
  • 3.1 Finite Commutative Ring Structure
  • 3.2 Regular Polynomials in the Ring R[x]
  • 3.3 R-algebra Automorphisms of R[x]
  • 3.4 Factorization in R[x]
  • 4 Separable Extensions of Finite Fields and Finite Rings
  • 4.1 Separable Field Extensions
  • 4.2 Extensions of Rings
  • 4.3 Separable extensions of finite commutative local rings
  • 5 Galois Theory for Local Rings
  • 5.1 Basic Facts
  • 5.2 Examples. Splitting Rings
  • 6 Galois and Quasi-Galois Rings: Structure and Properties
  • 6.1 Classical Constructions
  • 6.2 Galois Ring Properties
  • 6.3 Structure Theorems for finite commutative local rings
  • 6.4 Another class of finite commutative local rings: Quasi-Galois Rings
  • 7 Basic Notions on Codes over Finite Fields
  • 7.1 Basic properties
  • 7.2 Some families of q-ary codes
  • 7.3 Duality between codes
  • 7.4 Some families of nonlinear q-ary codes
  • 8 Basic Notions on Codes over Galois Rings
  • 8.1 Basic properties
  • 8.2 Linear quaternary codes
  • 8.3 Kerdock and Preparata codes revisited.

Similar Items