Finite Fields and Applications (Student Mathematical Library) by Gary L. Mullen (PDF)

Description

This book provides a brief and accessible introduction to the theory of finite fields and to some of their many fascinating and practical applications. The first chapter is devoted to the theory of finite fields. After covering their construction and elementary properties, the authors discuss the trace and norm functions, bases for finite fields, and properties of polynomials over finite fields. Each of the remaining chapters details applications. Chapter 2 deals with combinatorial topics such as the construction of sets of orthogonal latin squares, affine and projective planes, block designs, and Hadamard matrices. Chapters 3 and 4 provide a number of constructions and basic properties of error-correcting codes and cryptographic systems using finite fields. Each chapter includes a set of exercises of varying levels of difficulty which help to further explain and motivate the material. Appendix A provides a brief review of the basic number theory and abstract algebra used in the text,

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐To transmit photos from Mars back to earth, to download files from Internet where the server resides a few thousand miles away, and to send email globally are fascinating. Thus to understand the mathematics which made “… possible for the person receiving a message to detect and correct errors that have arisen during the transmission process” is interesting. “The detection of errors is accomplished by noticing that the received sequence is not a codeword … For some code, it is possible for the receiver to determine, with high probability, the intended message when the received sequence is not a codeword. Such codes are … called error-correcting codes.” Gary L. Mullen and Carl Mummert’s “Finite Field and Applications” introduces the error-correcting codes (algebraic coding theory) and the related mathematics. The book has four chapters. They are: finite fields, combinatorics, algebraic coding theory, and cryptography.The chapter of algebraic coding theory includes (a) basic properties of codes (linear code, parity-check digits, parity-check matrix, parity-check equation, and systematic form), (b) bounds for parameters of codes (Hamming bound, Plotkin bound, Singleton bound, and Gilbert-varshamov bound), (c) decoding methods (nearest neighbor decoding and syndrome decoding method), (d) code constructions (Hamming, cyclic, BCH, and Goppa codes), (e) codes and combinatorial designs, and (f) codes and Latin squares.On chapter four, symmetric key cryptography (automatic teller machines over public phone lines), public key cryptography (the RSA cryptosystem, Double-round quadratic enciphering, the Diffie-Hellman system, Elliptic curves and elliptic curve cryptography), and threshold scheme (to split a piece of secret information among several individuals) are introduced.The chapter of combinatorics introduces (a) Latin squares, (b) affine and projective planes, (c) block designs, and (d) Hadamard matrices.The first chapter, the most important chapter, introduces finite fields, extension fields, trace and norm functions, bases (linear algebra), and polynomials (over finite fields). It is interesting to learn the properties of finite fields such as: prime subfield and the field Fq is isomorphic to the ring Zp of integers modulo p (when p is a prime). The properties of the polynomials in finite field are also interesting. For example, “every function defined on a finite field can be represented by a polynomial with coefficients in that field.” In other words, “Every function f:Fq->Fq can be represented by a unique polynomial over Fq of degree at most q-1.” Polynomials have many important properties: (a) values are easily computable, (b) easily differentiated and integrated, and (c) vanish at only a finite number of points.

⭐This is a very good guide for new learners reading on finite fields

⭐

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