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 ininformation and communication engineering, for instance, in signal processing, coding theory, cryptography or computer science.

Autorentext

Dirk Hachenberger is a mathematician working in the fields of combinatorics, number theory, applicable algebra, finite geometry and coding theory. He is the recipient of the 2004 Hall medal of the Institute for Combinatorics and its Applications (ICA). He was also awarded the price for good teaching of the Bavarian State Minister for Science, Research and Arts in 2004. He has published the books "Finite Fields: Normal bases and completely free elements" (in English) and "Mathematics for computer scientists" (in German).

Dieter Jungnickel is an internationally known mathematician working in the fields of applicable algebra, coding theory, design theory, finite geometry, combinatorics and combinatorial optimization. He is the recipient of the 2018 Euler medal of the Institute for Combinatorics and its Applications (ICA). He has published several well-known books, including "Design Theory", "Optimization Methods", "Finite fields: Structure and arithmetics", "Coding Theory", "Combinatorics" and "Graphs, Networks and Algorithms", some of which have been published both in English and German.

Inhalt

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.