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＜E-Book＞
Topics in Galois Fields / by Dirk Hachenberger, Dieter Jungnickel
(Algorithms and Computation in Mathematics ; 29)

Edition	1st ed. 2020.
Publisher	(Cham : Springer International Publishing : Imprint: Springer)
Year	2020
Language	English
Size	XIV, 785 p. 11 illus : online resource
Authors	*Hachenberger, Dirk author Jungnickel, Dieter author SpringerLink (Online service)
Subjects	LCSH:Algebraic fields LCSH:Polynomials LCSH:Algebra LCSH:Number theory LCSH:Discrete mathematics LCSH:Computer science -- Mathematics  All Subject Search FREE:Field Theory and Polynomials FREE:Algebra FREE:Number Theory FREE:Discrete Mathematics FREE:Mathematics of Computing
Notes	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 ininformation and communication engineering, for instance, in signal processing, coding theory, cryptography or computer science HTTP:URL=https://doi.org/10.1007/978-3-030-60806-4

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