- 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.
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- Über dieses Lehrbuch
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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.
- Über die Autor*innen
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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.
- Inhaltsverzeichnis (14 Kapitel)
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Basic Algebraic Structures and Elementary Number Theory
Seiten 1-70
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Basics on Polynomials
Seiten 71-100
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Field Extensions and the Basic Theory of Galois Fields
Seiten 101-173
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The Algebraic Closure of a Galois Field
Seiten 175-196
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Irreducible Polynomials Over Finite Fields
Seiten 197-239
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Inhaltsverzeichnis (14 Kapitel)
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Bibliografische Information
- Bibliographic Information
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- Buchtitel
- Topics in Galois Fields
- Autoren
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- Dirk Hachenberger
- Dieter Jungnickel
- Titel der Buchreihe
- Algorithms and Computation in Mathematics
- Buchreihen Band
- 29
- Copyright
- 2020
- Verlag
- Springer International Publishing
- Copyright Inhaber
- Springer Nature Switzerland AG
- eBook ISBN
- 978-3-030-60806-4
- DOI
- 10.1007/978-3-030-60806-4
- Hardcover ISBN
- 978-3-030-60804-0
- Buchreihen ISSN
- 1431-1550
- Auflage
- 1
- Seitenzahl
- XIV, 785
- Anzahl der Bilder
- 11 schwarz-weiß Abbildungen
- Themen
