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Topics in Galois fields

Hachenberger, Dirk.

Topics in Galois fields

Record Type:	Electronic resources : Monograph/item
Title/Author:	Topics in Galois fieldsby Dirk Hachenberger, Dieter Jungnickel.
Author:	Hachenberger, Dirk.
other author:	Jungnickel, Dieter.
Published:	Cham :Springer International Publishing :2020.
Description:	xiv, 785 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Finite fields (Algebra)
Online resource:	https://doi.org/10.1007/978-3-030-60806-4
ISBN:	9783030608064$q(electronic bk.)

Topics in Galois fields
Hachenberger, Dirk.

Topics in Galois fields[electronic resource] /by Dirk Hachenberger, Dieter Jungnickel. - Cham :Springer International Publishing :2020. - xiv, 785 p. :ill., digital ;24 cm. - Algorithms and computation in mathematics,v.291431-1550 ;. - Algorithms and computation in mathematics ;v.5..

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.

ISBN: 9783030608064$q(electronic bk.)

Standard No.: 10.1007/978-3-030-60806-4doiSubjects--Topical Terms:

359950
Finite fields (Algebra)

LC Class. No.: QA247.3 / .H33 2020

Dewey Class. No.: 512.3

Topics in Galois fields
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490 1 $a Algorithms and computation in mathematics, $x 1431-1550 ; $v v.29
505 0 $a 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.
520 $a 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.
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