Preface; Elementary set theory and methods of proof; Numbers and polynomials; Binary relations an binary operations; Introduction to rings; Factor rings and fields; Basic group theory; Structure theorems of group theory; A brief excursion into Galois theory; Partial solutions; Bibliograpy; Notation; Index. The author provides a mixture of informal and formal material which help to stimulate the enthusiasm of the student, whilst still providing the essential theoretical concepts necessary for serious study. Retaining the highly readable style of its predecessor, this second edition has also been thoroughly revised to include a new chapter on Galois theory plus hints and solutions to many of the exercises featured. We are always looking for ways to improve customer experience on Elsevier. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit.
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It is a highly developed example of the power of generalisation and axiomatisation in mathematics. The first objective is to develop some concepts of set theory as a useful language for abstract mathematics. Next we aim to introduce the student to some of the basic techniques of algebra. Thirdly, we aim to introduce group theory through axioms and examples both because groups are typical and famous examples of algebraic structures, and because of their use in the measurement of symmetry.
Finally we begin the study of fields and rings which are important in the study of polynomials and their zeros. Learning Outcomes Students will have developed an abstract approach to reasoning about number systems, their arithmetic structures and properties.
They will be familiar with the axioms of a group, ring, and field together with examples. They will have a detailed knowledge of the ring of integers and the ring of polynomials and the Euclidean Algorithm in each case. They will have developed an appreciation of the isomorphism theorem for groups and rings. Concept of a binary operation. Axioms for a group. Axioms for a ring not assumed commutative with identity.
Definition of a field and an integral domain, with preliminary examples. Fact that a field is an integral domain and result that a finite integral domain is a field. Division algorithm and Euclidean algorithm. Statement of Fundamental Theorem of Arithmetic. Polynomials with real or complex coefficients; ring structure, degree of a polynomial; Division algorithm, Euclid's algorithm; the Remainder Theorem.
Groups in general, with additional straightforward examples. Subgroups, with examples; subgroup of a cyclic group is cyclic. Products of groups. Cosets and Lagrange's theorem; simple examples. The order of an element. Kernels, normal subgroups, The 1st Isomorphism Theorem. Please note that e-book versions of many books in the reading lists can be found on SOLO.
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Course Term:. Course Weight:. Course Overview:. Course Syllabus:. Reading List:. Herstein, An Introduction to Abstract Algebra. Alternative Reading N. Peter J. John B. Joseph J. Powered by Drupal.
Rings, Fields and Groups : An Introduction to Abstract Algebra, 2nd Edition
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Book:R.B.J.T. Allenby/Rings, Fields and Groups: An Introduction to Abstract Algebra/Second Edition
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An Introduction to Groups, Rings, and Fields (2009-2010)
Rings, Fields and Groups