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Solutions B Hints and Answers to Selected Exercises

Part I Groups

Chapter 1 Preliminaries

Section 1.2 Sets and Functions
Exercises 1.2.4 Practice Problems
Exercises 1.2.5 Collected Homework
Section 1.3 Equivalence Relations and Partitions
Exercises Practice Problems
Exercises 1.5 Additional Exercises
1.5.40. Fibonacci Numbers.

Chapter 2 Groups

Section 2.2 Definitions and Examples
Exercises 2.2.3 Practice Problems
Exercises 2.2.4 Collected Homework
Section 2.3 Basic Properties of Groups
Exercises Collected Homework
Section 2.4 Subgroups
Exercises 2.4.4 Practice Problems
Exercises 2.4.5 Collected Homework
Section 2.6 Permutation Groups
Exercises 2.6.7 Practice Problems
Exercises 2.7 Additional Exercises

Chapter 3 Cosets and Factor Groups

Section 3.1 Cosets
Exercises Practice Problems
Exercises Collected Homework
Section 3.2 Factor Groups and Normal Subgroups
Exercises 3.2.4 Practice Problems
Exercises 3.2.5 Collected Homework
Exercises 3.3 Exercises

Chapter 4 Isomorphisms and Homomorphisms

Section 4.1 Definition and Examples
Exercises 4.1.4 Practice Problems
Exercises 4.1.5 Collected Homework
Section 4.2 Group Homomorphisms
Exercises Collected Homework
Section 4.3 The Isomorphism Theorems
Exercises Practice Problems
Exercises 4.4 Additional Exercises

Part II Rings and Fields

Chapter 5 Rings

Section 5.1 Rings
Exercises Practice Problems
Section 5.2 Integral Domains and Fields
Exercises Collected Homework
Section 5.3 Ring Homomorphisms and Ideals
Exercises Practice Problems
Exercises Collected Homework
Exercises 5.6 Additional Exercises

Chapter 6 Polynomials

Section 6.1 The Integers
Exercises 6.1.5 Exercises
Section 6.3 The Division Algorithm
Exercises Practice Problems
Section 6.4 Irreducible Polynomials
Exercises 6.4.5 Collected Homework
Section 6.5 Factoring over \(\C\) and \(\R\)
Exercises 6.5.3 Practice Problems
Exercises 6.5.4 Collected Homework
Exercises 6.6 Additional Exercises
6.6.20. Cyclotomic Polynomials.

Chapter 7 Integral Domains

Exercises 7.4 Exercises

Chapter 8 Fields

Section 8.1 Geometric Constructions
Exercises 8.1.4 Collected Homework
Section 8.3 Extension Fields
Exercises 8.3.3 Collected Homework
Section 8.4 Algebraic Elements
Exercises Collected Homework
Section 8.7 What's Not Constructible
Exercises 8.7.4 Collected Homework

Chapter 9 Galois Theory

Section 9.1 Splitting Fields
Exercises 9.1.2 Collected Homework
Section 9.2 Field Automorphisms
Exercises Collected Homework

Part III More Group Theory

Chapter 10 Some of the Structure of Groups

Section 10.1 Isomorphisms and Cayley's Theorem
Exercises 10.1.4 Collected Homework
Section 10.2 Permutation Groups
Exercises 10.2.6 Practice Problems

Chapter 11 More of the Structure of Groups

Section 11.1 Cyclic Groups and Orders of Elements
Exercises 11.1.5 Collected Homework

Chapter 12 Introduction to Cryptography

Exercises 12.5 Exercises

Chapter 13 Group Actions

Section 13.3 Burnside's Counting Theorem
Exercises 13.3.3 Collected Homework
Exercises 13.4 Exercises

Chapter 14 The Sylow Theorems

Exercises 14.4 Exercises

Part IV Additional Topics

Chapter 15 Matrix Groups and Symmetry

Exercises 15.4 Exercises

Chapter 16 Algebraic Coding Theory

Exercises 16.6 Exercises

Chapter 17 Finite Fields

Exercises 17.4 Exercises
17.4.24. Wilson's Theorem.

Chapter 18 Lattices and Boolean Algebras

Exercises 18.5 Exercises