Overview
- Includes modern material that has not appeared in book form before.
- Develops the theory from the basics and includes a thorough discussion of infinite algebraic extensions of. Galois fields and generators thereof.
- Emphasizes (the concrete construction of) particular generators for Galois field extensions, which are motivated by several applications?.
Part of the book series: Algorithms and Computation in Mathematics (AACIM, volume 29)
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Table of contents (14 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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.
Reviews
“The book will appeal to specialists in finite field theory and to readers with a knowledge of modern algebra who would like to learn about finite field theory. In summary, this text does what it claims to do. It is a well-crafted text on finite field theory and a welcome addition to existing finite field theory literature.” (Charles Traina, MAA Reviews, June 13, 2021)
Authors and Affiliations
About the authors
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”, “CodingTheory”, “Combinatorics” and “Graphs, Networks and Algorithms”, some of which have been published both in English and German.
Bibliographic Information
Book Title: Topics in Galois Fields
Authors: Dirk Hachenberger, Dieter Jungnickel
Series Title: Algorithms and Computation in Mathematics
DOI: https://doi.org/10.1007/978-3-030-60806-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-60804-0Published: 30 September 2020
eBook ISBN: 978-3-030-60806-4Published: 29 September 2020
Series ISSN: 1431-1550
Edition Number: 1
Number of Pages: XIV, 785
Number of Illustrations: 11 b/w illustrations
Topics: Field Theory and Polynomials, Algebra, Number Theory, Combinatorics, Mathematics of Computing