A first course in abstract algebra
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Bibliographic Information
A first course in abstract algebra
Prentice Hall, c2000
2nd ed
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Note
Includes bibliographical references (p. 519-520) and index
Description and Table of Contents
Description
For one-semester or two-semester undergraduate courses in Abstract Algebra.
This new edition has been completely rewritten. The four chapters from the first edition are expanded, from 257 pages in first edition to 384 in the second. Two new chapters have been added: the first 3 chapters are a text for a one-semester course; the last 3 chapters are a text for a second semester. The new Chapter 5, Groups II, contains the fundamental theorem of finite abelian groups, the Sylow theorems, the Jordan-Holder theorem and solvable groups, and presentations of groups (including a careful construction of free groups). The new Chapter 6, Commutative Rings II, introduces prime and maximal ideals, unique factorization in polynomial rings in several variables, noetherian rings and the Hilbert basis theorem, affine varieties (including a proof of Hilbert's Nullstellensatz over the complex numbers and irreducible components), and Grobner bases, including the generalized division algorithm and Buchberger's algorithm.
Table of Contents
1. Number Theory.
Induction. Binomial Coefficients. Greatest Common Divisors. The Fundamental Theorem of Arithmetic. Congruences. Dates and Days.
2. Groups I.
Functions. Permutations. Groups. Lagrange's Theorem. Homomorphisms. Quotient Groups. Group Actions. Counting with Groups.
3. Commutative Rings I.
First Properties. Fields. Polynomials. Homomorphisms. Greatest Common Divisors. Unique Factorization. Irreducibility. Quotient Rings and Finite Fields. Officers, Fertilizer, and a Line at Infinity.
4. Goodies.
Linear Algebra. Euclidean Constructions. Classical Formulas. Insolvability of the General Quintic. Epilog.
5. Groups II.
Finite Abelian Groups. The Sylow Theorems. The Jordan-Hoelder Theorem. Presentations.
6. Commutative Rings II.
Prime Ideals and Maximal Ideals. Unique Factorization. Noetherian Rings. Varieties. Groebner Bases.
Hints to Exercises.
Bibliography.
Index.
by "Nielsen BookData"