Abstract Algebra: Theory and Applications

This is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. The first half of the book presents group theory, through the Sylow theorems. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galo

Abstract and Geometric Algebra Mathematics

Tác giả: Thomas W. Judson
Nhà xuất bản: Stephen F. Austing State University
Nguồn: LibreTexts
Giấy phép: GNU FDL
Ngôn ngữ: EN
Danh mục: Khoa học cơ bản

Đọc trực tuyến Tải PDF (nguồn mở)

Mục lục· 205 mục

01: Preliminaries
1.01: A Short Note on Proofs
1.02: Sets and Equivalence Relations
1.03: Reading Questions
1.04: Exercises
1.05: References and Suggested Readings
1.06: Sage
1.07: Sage Exercises
02: The Integers
2.01: Mathematical Induction
2.02: The Division Algorithm
2.03: Reading Questions
2.04: Exercises
2.05: Programming Exercises
2.06: References and Suggested Readings
2.07: Sage
2.08: Sage Exercises
03: Groups
3.01: Reading Questions
3.02: Exercises
3.03: Additional Exercises- Detecting Errors
3.04: References and Suggested Readings
3.05: Sage
3.06: Sage Exercises
3.07: Integer Equivalence Classes and Symmetries
3.08: Definitions and Examples
3.09: Subgroups
04: Cyclic Groups
4.01: Cyclic Subgroups
4.02: Multiplicative Group of Complex Numbers
4.03: The Method of Repeated Squares
4.04: Reading Questions
4.05: Exercises
4.06: Programming Exercises
4.07: References and Suggested Readings
4.08: Sage
4.09: Sage Exercises
05: Permutation Groups
5.01: Definitions and Notation
5.02: Dihedral Groups
5.03: Reading Questions
5.04: Exercises
5.05: Sage
5.06: Sage Exercises
06: Cosets and Lagrange's Theorem
6.01: Cosets
6.02: Lagrange's Theorem
6.03: Fermat's and Euler's Theorems
6.04: Reading Questions
6.05: Exercises
6.06: Sage
6.07: Sage Exercises
07: Introduction to Cryptography
7.01: Private Key Cryptography
7.02: Public Key Cryptography
7.03: Reading Questions
7.04: Exercises
7.05: Additional Exercises- Primality and Factoring
7.06: References and Suggested Readings
7.07: Sage
7.08: Sage Exercises
08: Algebraic Coding Theory
8.01: Sage Exercises
8.02: Error-Detecting and Correcting Codes
8.03: Linear Codes
8.04: Parity-Check and Generator Matrices
8.05: Efficient Decoding
8.06: Reading Questions
8.07: Exercises
8.08: Programming Exercises
8.09: References and Suggested Readings
8.10: Sage
09: Isomorphisms
9.01: Definition and Examples
9.02: Direct Products
9.03: Reading Questions
9.04: Exercises
9.05: Sage
9.06: Sage Exercises
10: Normal Subgroups and Factor Groups
10.01: Factor Groups and Normal Subgroups
10.02: The Simplicity of the Alternating Groups
10.03: Reading Questions
10.04: Exercises
10.05: Sage
10.06: Sage Exercises
11: Homomorphisms
11.01: Group Homomorphisms
11.02: The Isomorphism Theorms
11.03: Reading Questions
11.04: Exercises
11.05: Additional Exercises- Automorphisms
11.06: Sage
11.07: Sage Exercises
12: Matrix Groups and Symmetry
12.01: Matrix Groups
12.02: Symmetry
12.03: Reading Questions
12.04: Exercises
12.05: References and Suggested Readings
12.06: Sage
12.07: Sage Exercises
13: The Structure of Groups
13.01: Finite Abelian Groups
13.02: Solvable Groups
13.03: Reading Questions
13.04: Exercises
13.05: Programming Exercises
13.06: References and Suggested Readings
13.07: Sage
13.08: Sage Exercises
14: Group Actions
14.01: Groups Acting on Sets
14.02: The Class Equation
14.03: Burnside's Counting Theorem
14.04: Reading Questions
14.05: Exercises
14.06: Programming Exercise
14.07: References and Suggested Reading
14.08: Sage
14.09: Sage Exercises
15: The Sylow Theorems
15.01: The Sylow Theorems
15.02: Examples and Applications
15.03: Reading Questions
15.04: Exercises
15.05: A Project
15.06: References and Suggested Readings
15.07: Sage
15.08: Sage Exercises
16: Rings
16.02: Sage Exercises
16.03: Rings
16.04: Integral Domains and Fields
16.05: Ring Homomorphisms and Ideals
16.06: Maximal and Prime Ideals
16.07: An Application to Software Design
16.08: Reading Questions
16.09: Exercises
16.10: Programming Exercise
16.10: Sage
16.11: References and Suggested Readings
17: Polynomials
17.01: Polynomial Rings
17.02: The Division Algorithm
17.03: Irreducible Polynomials
17.04: Reading Questions
17.05: Exercises
17.06: Additional Exercises- Solving the Cubic and Quartic Equations
17.07: Sage
17.08: Sage Exercises
18: Integral Domains
18.01: Fields of Fractions
18.02: Factorization in Integral Domains
18.03: Reading Questions
18.04: Exercises
18.05: References and Suggested Readings
18.06: Sage
18.07: Sage Exercises
19: Lattices and Boolean Algebras
19.01: Lattices
19.02: Boolean Algebras
19.03: The Algebra of Electrical Circuits
19.04: Reading Questions
19.05: Exercises
19.06: Programming Exercises
19.07: References and Suggested Readings
19.08: Sage
19.09: Sage Exercises
20: Vector Spaces
20.01: Definitions and Examples
20.02: Subspaces
20.03: Linear Independence
20.04: Reading Questions
20.05: Exercises
20.06: References and Suggested Readings
20.07: Sage
20.08: Sage Exercises
21: Fields
21.01: Extension Fields
21.02: Splitting Fields
21.03: Geometric Constructions
21.04: Reading Questions
21.05: Exercises
21.06: References and Suggested Readings
21.07: Sage
21.08: Sage Exercise
22: Finite Fields
22.01: Structure of a Finite Field
22.02: Polynomial Codes
22.03: Reading Questions
22.04: Exercises
22.05: Additional Exercises- Error Correction for BCH Codes
22.06: References and Suggested Readings
22.07: Sage
22.08: Sage Exercises
23: Galois Theory
23.01: Field Automorphisms
23.02: The Fundamental Theorem
23.03: Applications
23.04: Reading Questions
23.05: Exercises
23.06: References and Suggested Readings
23.07: Sage
23.08: Sage Exercises