Basic Algebra [modern]

__Basic Algebra__ and __Advanced Algebra__ systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole. Key topics and features of __Advanced Algebra__: \*Topics build upon the linear algebra, group theory, factorization of ideals, structure of fields, Galois theory, and elementary theory of modules as developed in __Basic Algebra__ \*Chapters treat various topics in commutative and noncommutative algebra, providing introductions to the theory of associative algebras, homological algebra, algebraic number theory, and algebraic geometry \*Sections in two chapters relate the theory to the subject of Gr?bner bases, the foundation for handling systems of polynomial equations in computer applications \*Text emphasizes connections between algebra and other branches of mathematics, particularly topology and complex analysis \*Book carries on two prominent themes recurring in __Basic Algebra__: the analogy between integers and polynomials in one variable over a field, and the relationship between number theory and geometry \*Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems \*The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them; it includes blocks of problems that illuminate aspects of the text and introduce additional topics __Advanced Algebra__ presents its subject matter in a forward-looking way that takes into account the historical development of the subject. It is suitable as a text for the more advanced parts of a two-semester first-year graduate sequence in algebra. It requires of the reader only a familiarity with the topics developed in __Basic Algebra__. Basic Algebra And Advanced Algebra Systematically Develop Concepts And Tools In Algebra That Are Vital To Every Mathematician, Whether Pure Or Applied, Aspiring Or Established. Together, The Two Books Give The Reader A Global View Of Algebra And Its Role In Mathematics As A Whole. Key Topics And Features Of Advanced Algebra: *topics Build Upon The Linear Algebra, Group Theory, Factorization Of Ideals, Structure Of Fields, Galois Theory, And Elementary Theory Of Modules As Developed In Basic Algebra *chapters Treat Various Topics In Commutative And Noncommutative Algebra, Providing Introductions To The Theory Of Associative Algebras, Homological Algebra, Algebraic Number Theory, And Algebraic Geometry *sections In Two Chapters Relate The Theory To The Subject Of Gröbner Bases, The Foundation For Handling Systems Of Polynomial Equations In Computer Applications *text Emphasizes Connections Between Algebra And Other Branches Of Mathematics, Particularly Topology And Complex Analysis *book Carries On Two Prominent Themes Recurring In Basic Algebra: The Analogy Between Integers And Polynomials In One Variable Over A Field, And The Relationship Between Number Theory And Geometry *many Examples And Hundreds Of Problems Are Included, Along With Hints Or Complete Solutions For Most Of The Problems *the Exposition Proceeds From The Particular To The General, Often Providing Examples Well Before A Theory That Incorporates Them; It Includes Blocks Of Problems That Illuminate Aspects Of The Text And Introduce Additional Topics Advanced Algebra Presents Its Subject Matter In A Forward-looking Way That Takes Into Account The Historical Development Of The Subject. It Is Suitable As A Text For The More Advanced Parts Of A Two-semester First-year Graduate Sequence In Algebra. It Requires Of The Reader Only A Familiarity With The Topics Developed In Basic Algebra. Transition To Modern Number Theory -- Wedderburn–artin Ring Theory -- Brauer Group -- Homological Algebra -- Three Theorems In Algebraic Number Theory -- Reinterpretation With Adeles And Ideles -- Infinite Field Extensions -- Background For Algebraic Geometry -- The Number Theory Of Algebraic Curves -- Methods Of Algebraic Geometry. Anthony W. Knapp. Along With A Companion Volume, Basic Algebra Includes Bibliographical References (p. 713-716) And Indexes. Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole. Key topics and features of Basic Algebra: *Linear algebra and group theory build on each other continually *Chapters on modern algebra treat groups, rings, fields, modules, and Galois groups, with emphasis on methods of computation throughout *Three prominent themes recur and blend together at times: the analogy between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory and geometry *Many examples and hundreds of problems are included, along with a separate 90-page section giving hints or complete solutions for most of the problems *The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them; includes blocks of problems that introduce additional topics and applications for further study *Applications to science and engineering (e.g., the fast Fourier transform, the theory of error-correcting codes, the use of the Jordan canonical form in solving linear systems of ordinary differential equations, and constructions of interest in mathematical physics) appear in sequences of problems Basic Algebra presents the subject matter in a forward-looking way that takes into account its historical development. It is suitable as a text in a two-semester advanced undergraduate or first-year graduate sequence in algebra, possibly supplemented by some material from Advanced Algebra at the graduate level. It requires of the reader only familiarity with matrix algebra, an understanding of the geometry and reduction of linear equations, and an acquaintance with proofs Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Advanced Algebra includes chapters on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry. Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems. Together the two books give the reader a global view of algebra and its role in mathematics as a whole. Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Advanced Algebra includes chapters on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry. Together the two books give the reader a global view of algebra, its role in mathematics as a whole and are suitable as texts in a two-semester advanced undergraduate or first-year graduate sequence in algebra. cover-image-large 1 front-matter 2 fulltext_001 24 fulltext_002 56 fulltext_003 111 fulltext_004 139 fulltext_005 232 fulltext_006 268 fulltext_007 326 fulltext_008 390 fulltext_009 471 fulltext_010 567 back-matter 606