BASIC ABSTRACT ALGEBRA Second edition, P. B. Bhattacharya, S. K.Jam, S. R. Nagpaul

 BASIC ABSTRACT ALGEBRA Second edition, P. B. Bhattacharya, S. K.Jam, S. R. Nagpaul

   This book is intended for seniors and beginning graduate students. It isself-contained and covers the topics usually taught at this level.The book is divided into live parts . 

   Part I (Chapters 1.- 3)is a prerequisite for the rest of the book. It contains an informal introduc-(ion to sets, number systems, matrices, and determinants. Results provedin Chapter I include the Schröder-Bernstein theorem and the cardinalityof the set of real numbers. In Chapter 2, starting from the well-orderingprinciple of natural numbers, some important algebraic properties ofintegers have been proved. Chapter 3 deals with matrices and determinants.It is expected that students would already be familiar with most of thematerial in Part I before reaching their senior year. Therefore, it can becompleted rapidly, skipped altogether, or simply referred to as necessary.

    Part II (Chapters 4—8) deals with groups. Chapters 4 and 5 provide a foundation in the basic concepts in groups, including G-sets and theirapplications. Normal series, solvable groups, and the Jordan—Holdertheorem are given in Chapter 6. The simplicity of the alternating groupA,andthe nonsolvability of S,, n> 4,are proved in Chapter 7. Chapter 8contains the theorem on the decomposition of a finitely generated abeliangroup as a direct sum of cyclic groups, and the Sylow theorems. Theinvariants of a finite abelian group and the structure of groups of ordersp2, pq, where p, q are primes, are given as applications.

   Part III (Chapters 9— 14) deals with rings and modules. Chapters 9— IIcover the basic concepts of rings, illustrated by numerous examples,including prime ideals, maximal ideals, UFD, PID, and so forth. Chapter12 deals with the ring of fractions of a commutative ring with respect toa multiplicative set. Chapter 13 contains a systematic development ofintegers, starting from Peano's axioms. Chapter 14 is an introduction tomodules and vector spaces. Topics discussed include completely reduciblemodules, free modules, and rank. 

   Part IV (Chapters 15-18) is concerned with field theory. Chapters 15and 16 contain the usual material on algebraic extensions, includingexistence and uniqueness of algebraic closure, and normal and separableextensions. Chapter 17 gives the fundamental theorem of Galois theoryand its application to the fundamental theorem of algebra. Chapter 18gives applications of Galois theory to some classical problems in algebraand geometry.

   Part V (Chapters 19—21) covers sOme additional topics not usuallytaught at the undergraduate level. Chapter 19 deals with modules withchain conditions leading to the Wedderburn—Artin theorem for semi-simple artinian rings. Chapter 20 deals with the rank of a matrix over aPID through Smith normal form. Chapter 21 gives the structure of afinitely generated module over a ND and its applications to linear algebra.

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