Volume 69, Number 10, November 2022 Notices of the American Mathematical Society

Cover
Title page
Copyright
Contents
Preface
	For the student
	For the instructor
	Note about rings
	Road map
	Acknowledgments
Unit I: Preliminaries
Chapter 1. Introduction to Proofs
	1.1. Proving an implication
	1.2. Proof by cases
	1.3. Contrapositive
	1.4. Proof by contradiction
	1.5. If and only if
	1.6. Counterexample
	Exercises
Chapter 2. Sets and Subsets
	2.1. What is a set?
	2.2. Set of integers and its subsets
	2.3. Closure
	2.4. Showing set equality
	Exercises
Chapter 3. Divisors
	3.1. Divisor
	3.2. GCD theorem
	3.3. Proofs involving the GCD theorem
	Exercises
Unit II: Examples of Groups
Chapter 4. Modular Arithmetic
	4.1. Number system Z ₇
	4.2. Equality in Z ₇
	4.3. Multiplicative inverses
	Exercises
Chapter 5. Symmetries
	5.1. Symmetries of a square
	5.2. Group properties of ????₄
	5.3. Centralizer
	Exercises
Chapter 6. Permutations
	6.1. Permutations of the set {1,2,3}
	6.2. Group properties of ????_{????}
	6.3. Computations in ????_{????}
	6.4. Associative law in ????_{????} (and in ????_{????})
	Exercises
Chapter 7. Matrices
	7.1. Matrix arithmetic
	7.2. Matrix group ????(Z ₁₀)
	7.3. Multiplicative inverses
	7.4. Determinant
	Exercises
Unit III: Introduction to Groups
Chapter 8. Introduction to Groups
	8.1. Definition of a “group”
	8.2. Essential properties of a group
	8.3. Proving that a group is commutative
	8.4. Non-associative operations
	8.5. Direct product
	Exercises
Chapter 9. Groups of Small Size
	9.1. Smallest group
	9.2. Groups with two elements
	9.3. Groups with three elements
	9.4. Sudoku property
	9.5. Groups with four elements
	Exercises
Chapter 10. Matrix Groups
	10.1. Groups Z ₁₀ and ????₁₀
	10.2. Groups ????(Z ₁₀) and ????(Z ₁₀)
	10.3. Group ????(Z ₁₀)
	Exercises
Chapter 11. Subgroups
	11.1. Examples of subgroups
	11.2. Subgroup proofs
	11.3. Center and centralizer revisited
	Exercises
Chapter 12. Order of an Element
	12.1. Motivating example
	12.2. When does ????^{????}=?????
	12.3. Conjugates
	12.4. Order in an additive group
	12.5. Elements with infinite order
	Exercises
Chapter 13. Cyclic Groups, Part I
	13.1. Generators of the additive group Z ₁₂
	13.2. Generators of the multiplicative group ????₁₃
	13.3. Matching Z ₁₂ and ????₁₃
	13.4. Taking positive and negative powers of ????
	13.5. When the group operation is addition
	Exercises
Chapter 14. Cyclic Groups, Part II
	14.1. Why negative powers are needed
	14.2. Additive groups revisited
	14.3. ⟨3⟩ behaves “just like” Z
	14.4. Subgroups of cyclic groups
	Exercises
Unit IV: Group Homomorphisms
Chapter 15. Functions
	15.1. Domain and codomain
	15.2. One-to-one function
	15.3. Onto function
	15.4. When domain and codomain have the same size
	Exercises
Chapter 16. Isomorphisms
	16.1. Groups Z ₁₂ and ⟨????⟩: Elements match up
	16.2. Groups Z ₁₂ and ⟨????⟩: Operations match up
	16.3. Elements with infinite order revisited
	16.4. Inverse isomorphisms
	Exercises
Chapter 17. Homomorphisms, Part I
	17.1. Group homomorphism
	17.2. Properties of homomorphisms
	17.3. Order of an element
	Exercises
Chapter 18. Homomorphisms, Part II
	18.1. Kernel of a homomorphism
	18.2. Image of a homomorphism
	18.3. Partitioning the domain
	18.4. Finding homomorphisms
	Exercises
Unit V: Quotient Groups
Chapter 19. Introduction to Cosets
	19.1. Multiplicative group example
	19.2. Additive group example
	19.3. Right cosets
	19.4. Properties of cosets
	19.5. When are cosets equal?
	Exercises
Chapter 20. Lagrange’s Theorem
	20.1. Motivating Lagrange’s theorem
	20.2. Proving Lagrange’s theorem
	20.3. Applications of Lagrange’s theorem
	Exercises
Chapter 21. Multiplying/Adding Cosets
	21.1. Turning a set of cosets into a group
	21.2. Coset multiplication shortcut
	21.3. Cosets of ????=5Z in Z revisited
	Exercises
Chapter 22. Quotient Group Examples
	22.1. Quotient group ????₁₃/???? revisited
	22.2. Quotient group ????₃₇/????
	22.3. Quotient group ????/???? (generalization)
	Exercises
Chapter 23. Quotient Group Proofs
	23.1. Sample quotient group proofs
	23.2. Collapsing ???? into ????/????
	Exercises
Chapter 24. Normal Subgroups
	24.1. How does the shortcut fail and work?
	24.2. Normal subgroups: What and why
	24.3. Examples of normal subgroups
	24.4. Normal subgroup test
	Exercises
Chapter 25. First Isomorphism Theorem
	25.1. Familiar homomorphism
	25.2. Another homomorphism
	25.3. First Isomorphism Theorem
	25.4. Finding and building homomorphisms
	Exercises
Unit VI: Introduction to Rings
Chapter 26. Introduction to Rings
	26.1. Examples and definition
	26.2. Fundamental properties
	26.3. Units and zero divisors
	26.4. Subrings
	26.5. Group of units
	Exercises
Chapter 27. Integral Domains and Fields
	27.1. Integral domains
	27.2. Fields
	27.3. Idempotent elements
	Exercises
Chapter 28. Polynomial Rings, Part I
	28.1. Examples and definition
	28.2. Degree of a polynomial
	28.3. Units and zero divisors
	Exercises
Chapter 29. Polynomial Rings, Part II
	29.1. Division algorithm in ????[????]
	29.2. Factor theorem
	29.3. Nilpotent elements
	Big picture stuff
	Exercises
Chapter 30. Factoring Polynomials
	30.1. Examples and definition
	30.2. Factorable or unfactorable?
	Big picture stuff
	Exercises
Unit VII: Quotient Rings
Chapter 31. Ring Homomorphisms
	31.1. Evaluation map
	31.2. Properties of ring homomorphisms
	31.3. Kernel and image
	31.4. Examples and definition of an ideal
	31.5. Ideals in Z and in ????[????]
	Big picture stuff
	Exercises
Chapter 32. Introduction to Quotient Rings
	32.1. From a quotient group to a quotient ring
	32.2. Role of an ideal in a quotient ring
	32.3. Quotient ring Z ₃[????]/⟨????²⟩
	32.4. First Isomorphism Theorem for rings
	Big picture stuff
	Exercises
Chapter 33. Quotient Ring Z ₇[????]/⟨????²-1⟩
	33.1. Division algorithm revisited
	33.2. Another way to reduce in Z ₇[????]/⟨????²-1⟩
	33.3. ????[????]/⟨????(????)⟩ is not a field
	33.4. ????[????]/⟨????(????)⟩ is a field
	Big picture stuff
	Exercises
Chapter 34. Quotient Ring R [????]/⟨????²+1⟩
	34.1. Reducing elements in R [????]/⟨????²+1⟩
	34.2. Field of complex numbers
	34.3. ????[????]/⟨????(????)⟩ is a field revisited
	Exercises
Chapter 35. ????[????]/⟨????(????)⟩ Is/Isn’t a Field, Part I
	35.1. Translate from ????[????] to Z
	35.2. Translate (back) from Z to ????[????]
	35.3. Proof of Theorem 35.1(b)
	Big picture stuff
	Exercises
Chapter 36. Maximal Ideals
	36.1. Examples and definition
	36.2. Maximality of ⟨????(????)⟩
	Big picture stuff
	Exercises
Chapter 37. ????[????]/⟨????(????)⟩ Is/Isn’t a Field, Part II
	37.1. Maximal ideals and quotient rings
	37.2. Putting it all together
	37.3. Oh wait, but there’s more!
	37.4. Prime ideals
	Exercises
Appendix A. Proof of the GCD Theorem
Appendix B. Composition Table for ????₄
Appendix C. Symbols and Notations
Appendix D. Essential Theorems
Index of Terms
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