Algebra 1. Groups, rings, fields and arithmetic (Q526644)

The book under review is the first part of a new, three-volume textbook on abstract algebra written by Professor Ramji Lal from the Harish Chandra Research Institute (HRI) in Allahabad, Uttar Pradesh, India. As the author points out in the preface to the present book, the aim of this series of three volumes is to develop various of the main principles of modern higher algebra for a wide audience of students, thereby starting from the undergraduate level and leading up to the advanced postgraduate level, simultaneously providing a very rich supply of instructive illustations, concrete examples, and related exercises. The (first) volume at issue is devoted to the introduction of the basic algebraic structures such as groups, rings, polynoriial rings, and fields, together with some elementary number theory and arithmetic in rings. This program is carried out in eleven chapters, the contents of which are organized as follows. Chapter 1 briefly provides the logical foundation of mathematical reasoning and proving, whereas Chapter 2 gives a first introduction to basic set theory as part of the general language of mathematics. In this context, the Zermelo-Fraenkel axiomatic system of set theory is carefully explained and, apart from the usual standard concepts, the axiom of choice as well as ordinal and cardinal numbers are discussed in a very detailed and rigorous way, with full proofs and abstract consequences. Chapter 3 introduces the number system, including the natural numbers, the integers, the rational numbers, the real numbers and the complex numbers. As for the treatment of the integer some of their elementary arithmetic is already developed and applied in this place, in particular the fundamental theorem of arithmetic, linear congruences, and rings of residue classes modulo an integer. Chapter 4 presents the basic concepts on groups and their homomorphisms, including subgroups, generators, and cyclic groups. This is continued in Chapter 5, where the coset decomposition of a group with respect to a subgroup, Lagrange's theorem, products of groups, quotient groups, and the Noether isomorphism theorems are dealt with. Chapter 6 turns to permutation groups, alternating maps, and the structure of alternating groups, together with the first elementary examples and properties of matrix groups, especially with a view toward general linear groups and some further classical groups. Thereafter, the elementary theory of rings and fields is developed in Chapter 7, with particular emphasis on ideals and polynomial rinds. The subsequent Chapter 8 returns to elementary number theory and, now with more algebraic tools being available, studies arithmetic functions, quadratic residues, the quadratic reciprocity law, and some special nonlinear Diophantine equations. Chapter 9 continues with the higher structure theory of groups, with the focus on group actions, the Sylow theorems, the classification of finite abelian groups, norn series and composition series of groups, and the Jordan-Hölder theorem. More structure theory of groups follows in Chapter 10, where the Remak-Krull-Schmidt theorem on direct decompositions of groups, the struct of solvable and nilpotent groups as well as free groups and presentations of groups are thoroughly described. The main text of this volume concludes with Chapter 11 which is devoted to arithmetical properties of rings. Division in rings, principal ideal domains, Euclidean domains, unique factorization domains, and the Chinese remainder theorem in rings are the main topics discussed in this chapter. Finally, there is a short appendix on the basic notions in the theory of categories and functors. The second volume [Algebra 2. Linear algebra, Galois theory, representation theory, group extensions and Schur multiplier. Singapore: Springer (2017; Zbl 1369.00003)] has appeared simultaneously and will be reviewed separately in the sequel. Its main topics are: the fundamentals of linear algebra, module theory, structure theory of fields and Galois theory, representation theory of finite groups, and the theory of group extensions. The planned third volume will be more advanced, including more commutative algebra, basic algebraic geometry, homological algebra, semi-simple Lie algebras, and Chevalley groups. The present first volume under review clearly shows the author's rich teaching experience of over 40 years. The text and its presentation are utmost carefully thought out, apparently well-tried in various algebra courses and written up in highly polished form. The rich material is explaind in a very lucid, detailed and rigorous style. Together with the unique wealth of instructive illustrations, examples, and exercises, the main text makes an excellent source for both teaching and self-study of the subject. Certainly, this new textbook is a very useful and valuable enrichment of the plenty of existing primers in the field of higher algebra.