The book provides an introduction to modern abstract algebra and its applications. It covers all major topics of classical theory of numbers, groups, rings, fields and finite dimensional algebras. The book also provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. In particular, it considers algorithm RSA, secret sharing algorithms, Diffie-Hellman Scheme and ElGamal cryptosystem based on discrete logarithm problem. It also presents Buchberger’s algorithm which is one of the important algorithms for constructing Gröbner basis.
- Covers all major topics of classical theory of modern abstract algebra such as groups, rings and fields and their applications. In addition it provides the introduction to the number theory, theory of finite fields, finite dimensional algebras and their applications.
- Provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics.
- Presents numerous examples illustrating the theory and applications. It is also filled with a number of exercises of various difficulty.
- Describes in detail the construction of the Cayley-Dickson construction for finite dimensional algebras, in particular, algebras of quaternions and octonions and gives their applications in the number theory and computer graphics.
Table of Contents
Introduction. Elements of Number Theory. Elements of the Theory of Groups. Examples of Groups. Elements of the Ring Theory. Polynomial Rings. Elements of the Theory of Fields. Examples of Applications. Polynomials in Several Variables. Finite Fields and their Applications. Finite Dimensional Algebras. Applications of Quarternions and Octionions.
Nadiya Gubareni, Associate Professor of Silesian University of Technology, Poland