LEADER 04016nam 22007575 450 001 9910483665103321 005 20250424094701.0 010 $a3-030-60806-9 024 7 $a10.1007/978-3-030-60806-4 035 $a(CKB)5590000000002295 035 $a(MiAaPQ)EBC6361021 035 $a(DE-He213)978-3-030-60806-4 035 $a(MiAaPQ)EBC6647501 035 $a(Au-PeEL)EBL6361021 035 $a(OCoLC)1198559148 035 $a(Au-PeEL)EBL6647501 035 $a(PPN)250220962 035 $a(EXLCZ)995590000000002295 100 $a20200929d2020 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aTopics in Galois Fields /$fby Dirk Hachenberger, Dieter Jungnickel 205 $a1st ed. 2020. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2020. 215 $a1 online resource (XIV, 785 p. 11 illus.) 225 1 $aAlgorithms and Computation in Mathematics,$x2512-3254 ;$v29 311 08$a3-030-60804-2 320 $aIncludes bibliographical references and index. 327 $aBasic 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. 330 $aThis monograph provides a self-contained presentation of the foundations of finite fields, including a detailed treatment of their algebraic closures. It also covers important advanced topics which are not yet found in textbooks: the primitive normal basis theorem, the existence of primitive elements in affine hyperplanes, and the Niederreiter method for factoring polynomials over finite fields. We give streamlined and/or clearer proofs for many fundamental results and treat some classical material in an innovative manner. In particular, we emphasize the interplay between arithmetical and structural results, and we introduce Berlekamp algebras in a novel way which provides a deeper understanding of Berlekamp's celebrated factorization algorithm. The book provides a thorough grounding in finite field theory for graduate students and researchers in mathematics. In view of its emphasis on applicable and computational aspects, it is also useful for readers working ininformation and communication engineering, for instance, in signal processing, coding theory, cryptography or computer science. 410 0$aAlgorithms and Computation in Mathematics,$x2512-3254 ;$v29 606 $aAlgebraic fields 606 $aPolynomials 606 $aAlgebra 606 $aNumber theory 606 $aDiscrete mathematics 606 $aComputer science$xMathematics 606 $aField Theory and Polynomials 606 $aAlgebra 606 $aNumber Theory 606 $aDiscrete Mathematics 606 $aMathematics of Computing 615 0$aAlgebraic fields. 615 0$aPolynomials. 615 0$aAlgebra. 615 0$aNumber theory. 615 0$aDiscrete mathematics. 615 0$aComputer science$xMathematics. 615 14$aField Theory and Polynomials. 615 24$aAlgebra. 615 24$aNumber Theory. 615 24$aDiscrete Mathematics. 615 24$aMathematics of Computing. 676 $a512.3 700 $aHachenberger$b Dirk$0845494 702 $aJungnickel$b D.$f1952- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910483665103321 996 $aTopics in Galois Fields$91887565 997 $aUNINA