Topics in Galois Fields

✍ Scribed by Dirk Hachenberger, Dieter Jungnickel

• Publisher: Springer International Publishing;Springer
• Year: 2020
• Tongue: English
• Leaves: 785
• Series: Algorithms and Computation in Mathematics 29
• Edition: 1st ed.
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter ....Pages i-xiv
Basic Algebraic Structures and Elementary Number Theory (Dirk Hachenberger, Dieter Jungnickel)....Pages 1-70
Basics on Polynomials (Dirk Hachenberger, Dieter Jungnickel)....Pages 71-100
Field Extensions and the Basic Theory of Galois Fields (Dirk Hachenberger, Dieter Jungnickel)....Pages 101-173
The Algebraic Closure of a Galois Field (Dirk Hachenberger, Dieter Jungnickel)....Pages 175-196
Irreducible Polynomials Over Finite Fields (Dirk Hachenberger, Dieter Jungnickel)....Pages 197-239
Factorization of Univariate Polynomials over Finite Fields (Dirk Hachenberger, Dieter Jungnickel)....Pages 241-295
Matrices Over Finite Fields (Dirk Hachenberger, Dieter Jungnickel)....Pages 297-353
Basis Representations and Arithmetics (Dirk Hachenberger, Dieter Jungnickel)....Pages 355-425
Shift Register Sequences (Dirk Hachenberger, Dieter Jungnickel)....Pages 427-487
Characters, Gauss Sums, and the DFT (Dirk Hachenberger, Dieter Jungnickel)....Pages 489-533
Normal Bases and Cyclotomic Modules (Dirk Hachenberger, Dieter Jungnickel)....Pages 535-579
Complete Normal Bases and Generalized Cyclotomic Modules (Dirk Hachenberger, Dieter Jungnickel)....Pages 581-621
Primitive Normal Bases (Dirk Hachenberger, Dieter Jungnickel)....Pages 623-687
Primitive Elements in Affine Hyperplanes (Dirk Hachenberger, Dieter Jungnickel)....Pages 689-743
Back Matter ....Pages 745-785

✦ Subjects

Mathematics; Field Theory and Polynomials; Algebra; Number Theory; Combinatorics; Mathematics of Computing