The book Algebra provides a firm foundation in algebra for students at undergraduate and postgraduate level.Starting with an introduction to Elementary Number Theory, the text gives a streamlined account of Group Theory, Ring Theory and Field Theory. The discussion on elementary number theory serves as a gentle introduction to the art of writing proofs and abstraction. The approach to topics such as symmetric groups and dihedral groups will be novel to the undergraduate students. The topic on Group Action emphasizes geometric intuition and it plays an important role. The idea of factorization, a recurring theme in rings is emphasized and done in detail. Two outstanding results in Field Theory, namely Galois Theorem and Abel's Theorem are proved efficiently.The book contains a wealth of examples and exercises with varying level of difficulty-quite a few of them drawn from other branches of mathematics. The text emphasizes on concrete mathematics.SOME KEY FEATURES:* Concrete examples precede abstract definitions* Elementary proofs of major results and their typical applications* Efficient proofs of Galois and Abel's Theorems
G. SanthanamDepartment of Mathematics & StatisticsIndian Institute of TechnologyKanpur
ELEMENTARY NUMBER THEORY: Divisibility / Diophantine Equations / Congruences / Chinese Remainder Theorem / Miscellaneous Exercises / GROUP THEORY I: Introduction to Groups / Subgroups / Lagrange's Theorem / Subgroups-continued / Homomorphisms / Product Groups / Quotient Groups / Special Class of Groups / RING THEORY: Definition and Examples / Divisibility in rings / Polynomials, Power Series, Laurent Series / Subrings and Ideals / Quotient Rings / Homomorphisms / Chinese Remainder Theorem / Field of fractions / Factorization / Group Rings / FIELD THEORY: Field Extensions / Galois and Abel's Theorems / Solvability of Cubic and Quartic polynomials / Constructions with straightedge and compass / Finite Fields / GROUP THEORY II: Group Action / Cauchy's Theorem / Sylow Theorems and applications / Finitely generated Abelian groups / Finite subgroups of E(n) / Solvable Groups.