Topics in Galois Fields /

Basic Algebraic Structures and Elementary Number Theory
Basics on Polynomials- Field Extensions and the Basic Theory of Galois Fields
The Algebraic Closure of a Galois Field
Irreducible Polynomials over Finite Fields
Factorization of Univariate Polynomials over Finite Fields
Matrices over Finite Fields
Basis Representations and Arithmetics
Shift Register Sequences
Characters, Gauss Sums, and the DFT
Normal Bases and Cyclotomic Modules
Complete Normal Bases and Generalized Cyclotomic Modules
Primitive Normal Bases
Primitive Elements in Affin Hyperplanes
List of Symbols
References
Index.

This monograph provides a self-contained presentation of the foundations of finite fields, including a detailed treatment of their algebraic closures. It also covers important advanced topics which are not yet found in textbooks: the primitive normal basis theorem, the existence of primitive elements in affine hyperplanes, and the Niederreiter method for factoring polynomials over finite fields.

We give streamlined and/or clearer proofs for many fundamental results and treat some classical material in an innovative manner. In particular, we emphasize the interplay between arithmetical and structural results, and we introduce Berlekamp algebras in a novel way which provides a deeper understanding of Berlekamp's celebrated factorization algorithm.

The book provides a thorough grounding in finite field theory for graduate students and researchers in mathematics. In view of its emphasis on applicable and computational aspects, it is also useful for readers working in information and communication engineering, for instance, in signal processing, coding theory, cryptography or computer science.