Algebra (2nd Edition) Solution by Michael Artin

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Algebra (2nd Edition) Solution written by Michael Artin , Massachusetts Institute of Technology.
According to Michael Artin:
This book began many years ago in the form of supplementary notes for my algebra classes. I wanted to discuss someconcretetopics such as symmetry, linear groups, and quadratic number fieldsin more detail than the text provided, and to shift the emphasis in group theory from permutation groupsto matrix groups. J...Jattices, another recurring theme, appeared spontaneously.
My hope was that the concrete material would interest the students and that it would make the abstractions more understandable- in short, that they could get farther bylearning both at the same time. This worked pretty welLIt tookmequite a while to decide what to include, but I gradually handed out more notesand eventually began teaching from them without another text. Though this produced a bookthat is different from most others, the problemsI encountered while fitting the parts together caused me many headaches. I can't recommend the method.

Algebra (2nd Edition) Solution written by Michael Artin cover the following topics.

1. Matrices
1.1 The Basic Operations
1.2 Row Reduction
1.3 The Matrix Transpose
1.4 Determinants
1.5 Permutations
1.6 Other Formulas for the Determinant
Exercises

2. Groups
2.1 Lawsof Composition
2.2 Groups and Subgroups
2.3 Subgroups of the Additive Group of Integers
2.4 Cyclic Groups
2.5 Homomorphisms
2.6 Isomorphisms
2.7 Equivalence Relations and Partitions
2.8 Cosets
2.9 Modular Arithmetic
2.10 The Correspondence Theorem
2.11 Product Groups
2.12 Quotient Groups
Exercises

3. Vector Spaces
3.1 Subspacesof }Rn
3.2 Fields
3.3 Vector Spaces
3.4 Bases and Dimension
3.5 Computing with Bases
3.6 Direct Sums
3.7 Infinite-Dimensional Spaces
Exercises

4. Linear Operators
4.1 The Dimension Formula
4.2 The Matrix of a Linear Transformation)
4.3 Linear Operators
4.4 Eigenvectors
4.5 The Characteristic Polynomial
4.6 Triangular and Diagonal Forms
4.7 JordanForm
Exercises

5. Applications of LinearOperators
5.1 Orthogonal Matrices and Rotations
5.2 Using Continuity
5.3 Systems of Differential Equations
5.4 The Matrix Exponential
Exercises

6. Symmetry
6.1 Symmetry of Plane Figures
6.2 Isometries
6.3 Isometries of the Plane
6.4 Finite Groups of Orthogonal Operators on the Plane
6.5 DiscreteGroups of Isometries
6.6 Plane Crystallographic Groups
6.7 Abstract Symmetry: Group Operations
6.8 TheOperation on Cosets . . . .
6.9 The Counting Formula ....
6.10 Operations on Subsets . . . .. ....
6.11 Permutation Representations . . . . . . . . . .
6.12 Finite Subgroups of the Rotation Group Exercises . . . . . . . . . . . . . . . . . .)

7. More Group Theory
7.1 Cayley's Theorem
7.2 The Class Equation
7.3 p-Groups
7.4 The Class Equation of the Icosahedral Group
7.5 Conjugation in the Symmetric Group
7.6 Normalizers
7.7 The Sylow Theorems
7.8 Groups of Order 12
7.9 The Free Group
7.10 Generators and Relations
7.11 The Todd-Coxeter Algorithm
Exercises

8. Bilinear Forms
8.1 Bilinear Forms
8.2 Symmetric Forms
8.3 Hermitian Forms
8.4 Orthogonality
8.5 Euclidean Spacesand Hermitian Spaces
8.6 The Spectral Theorem
8.7 Conics and Quadrics
8.8 Skew-Symmetric Forms
8.9 Summary
Exercises

9. Linear Groups
9.1 The ClassicalGroups
9.2 Interlude: Spheres
9.3 The Special Unitary Group SU2
9.4 The Rotation Group S03
9.5 One-Parameter Groups
9.6 The Lie Algebra
9.7 Translation in a Group
9.8 Normal Subgroups of SL2
Exercises

10. Group Representations
10.1 Definitions
10.2 Irreducible Representations
10.3 U ni tary Represen tations
10.4 Characters
10.5 One-DimensionalCharacters
10.6 TheRegularRepresentation
10.7 Schur
s Lemma
10.8 Proof of the Orthogonality Relations
10.9 Representations of SU2
Exercises

11. Rings
11.1 Definition of a Ring
11.2 Polynomial Rings
11.3 Homomorphisms and Ideals
11.4Quotient Rings
11.5 Adjoining Elements
11.6 Product Rings
11.7 Fractions
11.8Maximal Ideals
11.9 Algebraic Geometry
Exercises

12. Factoring
12.1 Factoring Integers
12.2 Unique Factorization Domains
12.3 Gauss
8 Lemma
12.4 Factoring Integer Polynomials
12.5 Gauss Primes
Exercises

13. Quadratic Number Fields
13.1 Algebraic Integers
13.2 Factoring Algebraic Integers
13.3 Ideals in Z[H ]
13.4 Ideal Multiplication
13.5 Factoring Ideals
13.6 Prime Ideals and Prime Integers
13.7 Ideal Classes
13.8 Computing the Class Group
13.9 Real Quadratic Fields
13.1 0 About Lattices
Exercises

14. Linear Algebra in a Ring
14.1 Modules
14.2Free Modules
14.3 Identi ti es
14.4 Diagonalizing Integer Matrices
14.5 Generators and Relations
14.6 Noetherian Rings
14.7 Structure of Abelian Groups
14.8 Application to Linear Operators
14.9 Polynomial Rings in Several Variables
Exercises

15. Fields
15.1 Examples of Fields
15.2 Algebraic and Transcendental Elements
15.3The Degree of a Field Extension
15.4 Finding the Irreducible Polynomial
15.5Ruler and Compass Constructions
15..6 Adjoining Roots
15.7 Finite Fields
15.8 Primitive Elements
15.9 Function Fields
15.10 The Fundamental Theorem of Algebra
Exercises

16. Galois Theory
16.1 SymmetricFunctions
16.2 The Discriminant
16.3 Splitting Fields
16.4 Isomorphisms of Field Extensions
16.5 Fixed Fields
16.6 Galois Extensions
16.7 The Main Theorem
16.8 Cubic Equations
16.9 Quartic Equations
16.10 Roots of Unity
16.11 Kummer Extensions
16.12 Quintic Equations
Exercises

APPENDIX (Background Material)
A.l About Proofs
A.2 The Integers
A.3 Zorn's Lemma
A.4 TheImplicit Function Theorem
Exercises

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