Algebra

by Artin, Michael

• ISBN13: 9780130047632
• ISBN10: 0130047635
• Edition: 1st
• Format: Hardcover
• Copyright: 1991-01-01
• Publisher: Addison Wesley

Summary

M->CREATED

Table of Contents

Preface	xiii
A Note for the Teacher	xv
Matrix Operations	1	(37)
The Basic Operations	1	(8)
Row Reduction	9	(9)
Determinants	18	(6)
Permutation Matrices	24	(4)
Cramer's Rule	28	(10)
Exercises	31	(7)
Groups	38	(40)
The Definition of a Group	38	(6)
Subgroups	44	(4)
Isomorphisms	48	(3)
Homomorphisms	51	(2)
Equivalence Relations and Partitions	53	(4)
Cosets	57	(2)
Restriction of a Homomorphism to a Subgroup	59	(2)
Products of Groups	61	(3)
Modular Arithmetic	64	(2)
Quotient Groups	66	(12)
Exercises	69	(9)
Vector Spaces	78	(31)
Real Vector Spaces	78	(4)
Abstract Fields	82	(5)
Bases of Dimension	87	(7)
Computation with Bases	94	(6)
Infinite-Dimensional Spaces	100	(2)
Direct Sums	102	(7)
Exercises	104	(5)
Linear Transformations	109	(46)
The Dimension Formula	109	(2)
The Matrix of a Linear Transformation	111	(4)
Linear Operators and Eigenvectors	115	(5)
The Characteristic Polynomial	120	(3)
Orthogonal Matrices and Rotations	123	(7)
Diagonalization	130	(3)
Systems of Differential Equations	133	(5)
The Matrix Exponential	138	(17)
Exercises	145	(10)
Symmetry	155	(42)
Symmetry of Plane Figures	155	(2)
The Group of Motions of the Plane	157	(5)
Finite Groups of Motions	162	(4)
Discrete Groups of Motions	166	(9)
Abstract Symmetry: Group Operations	175	(3)
The Operation on Cosets	178	(2)
The Counting Formula	180	(2)
Permutation Representations	182	(2)
Finite Subgroups of the Rotation Group	184	(13)
Exercises	188	(9)
More Group Theory	197	(40)
The Operations of a Group on Itself	197	(3)
The Class Equation of the Icosahedral Group	200	(3)
Operations on Subsets	203	(2)
The Sylow Theorems	205	(4)
The Groups of Order 12	209	(2)
Computation in the Symmetric Group	211	(6)
The Free Group	217	(2)
Generators and Relations	219	(4)
The Todd--Coxeter Algorithm	223	(14)
Exercises	229	(8)
Bilinear Forms	237	(33)
Definition of Bilinear Form	237	(6)
Symmetric Forms: Orthogonality	243	(4)
The Geometry Associated to a Positive Form	247	(2)
Hermitian Forms	249	(4)
The Spectral Theorem	253	(2)
Conics and Quadrics	255	(4)
The Spectral Theorem for Normal Operators	259	(1)
Skew-Symmetric Forms	260	(1)
Summary of Results, in Matrix Notation	261	(9)
Exercises	262	(8)
Linear Groups	270	(37)
The Classical Linear Groups	270	(2)
The Special Unitary Group SU2	272	(4)
The Orthogonal Representation of SU2	276	(5)
The Special Linear Group SL2(R)	281	(2)
One-Parameter Subgroups	283	(3)
The Lie Algebra	286	(6)
Translation in a Group	292	(3)
Simple Groups	295	(12)
Exercises	300	(7)
Group Representations	307	(38)
Difinition of a Group Representation	307	(3)
G-Invariant Forms and Unitary Representations	310	(2)
Compact Groups	312	(2)
G-Invariant Subspaces and Irreducible Representations	314	(2)
Characters	316	(5)
Permutation Representations and the Regular Representation	321	(2)
The Reprentations of the Icosahedral Group	323	(2)
One-Dimensional Representations	325	(1)
Schur's Lemma, and Proof of the Orthogonality Relations	325	(5)
Representations of the Group SU2	330	(15)
Exercises	335	(10)
Rings	345	(44)
Definition of a Ring	345	(2)
Formal Construction of Integers and Polynomials	347	(6)
Homomorphisms and Ideals	353	(6)
Quotient Rings and Relations in a Ring	359	(5)
Adjunction of Elements	364	(4)
Integral Domains and Fraction Fields	368	(2)
Maximal Ideals	370	(3)
Algebraic Geometry	373	(16)
Exercises	379	(10)
Factorization	389	(61)
Factorization of Integers and Polynomials	389	(3)
Unique Factorization Domains, Principal Ideal Domains, and Euclidean Domains	392	(6)
Gauss's Lemma	398	(4)
Explicit Factorization of Polynomials	402	(4)
Primes in the Ring of Gauss Integers	406	(3)
Algebraic Integers	409	(5)
Factorization in Imaginary Quadratic Fields	414	(5)
Ideal Factorization	419	(5)
The Relation Between Prime Ideals of R and Prime Integers	424	(1)
Ideal Classes in Imaginary Quadratic Fields	425	(8)
Real Qudratic Fields	433	(4)
Some Diophantine Equations	437	(13)
Exercises	440	(10)
Modules	450	(42)
The Definition of a Module	450	(2)
Matrices, Free Modules, and Bases	452	(4)
The Principle of Permanence of Identities	456	(1)
Diagonalization of Integer Matrices	457	(7)
Generators and Relations for Modules	464	(7)
The Structure Theorem for Abelian Groups	471	(5)
Application to Linear Operators	476	(6)
Free Modules over Polynomial Rings	482	(10)
Exercises	483	(9)
Fields	492	(45)
Examples of Fields	492	(1)
Algebraic and Transcendental Elements	493	(3)
The Degree of a Field Extension	496	(4)
Constructions with Ruler and Compass	500	(6)
Symbolic Adjunction of Roots	506	(3)
Finite Fields	509	(6)
Function Fields	515	(10)
Transcendental Extensions	525	(2)
Algebraically Closed Fields	527	(10)
Exercises	530	(7)
Galois Theory	537	(48)
The Main Theorem of Galois Theory	537	(6)
Cubic Equations	543	(4)
Symmetric Functions	547	(5)
Primitive Elements	552	(4)
Proof of the Main Theorem	556	(4)
Quartic Equations	560	(5)
Kummer Extensions	565	(2)
Cyclotomic Extensions	567	(3)
Quintic Equations	570	(15)
Exercises	575	(10)
Appendix Background Material	585	(16)
1. Set Theory	585	(4)
2. Techniques of Proof	589	(4)
3. Topology	593	(4)
4. The Implicit Function Theorem	597	(4)
Exercises	599	(2)
Notation	601	(2)
Suggestions for Further Reading	603	(4)
Index	607