## Calendar:

• 6-24-21: Final Exam
• 6-23-21: Review
• 6-22-21: Review Exam 3 and Make Up Exam
• 6-21-21: Exam 3
• 6-17-21: Kernels and the First isomorphism Theorem for Groups and Cauchy’s Theorem
• 6-16-21: Cosets and Lagrange’s Theorem, Normal Subgroups and Quotient Groups
• 6-15-21: Cayley’s Theorem and intro to homomorphisms/isomorphisms, Cosets and Subgroups
• 6-14-21: We went over problems 7, 8, 9, and 10 on Exam 2 with a strong emphasis on the importance of knowing and understanding definitions. We then spent time discussing symmetric groups and their connection to other groups such as the dihedral groups. Tomorrow we will generalize and formalize this with Cayley’s Theorem.
• 6-10-21: Exam 2
• 6-9-21: We discussed groups and subgroups and briefly mentioned cosets. We also looked at Cayley Graphs as a way to visualize groups. We went through some theorems about properties of groups and orders of elements.
• 6-8-21: Polished off some miscellaneous details about rings and ideals. Started talking about groups.
• 6-7-21: Today we went over question 6, 8, 10 and 11 on the exam. Then we talked about some results with congruences and polynomials (Theorem 5.10), introduced ideals in rings, and kernels of homomorphisms. We ended by proving the First Isomorphism Theorem. Tomorrow we will clean up a couple outstanding issues with rings and start groups.
• 6-3-21: Exam I is today, in preparation be sure you looked at the Unit 1 study guide below and looked at the sample exam at the bottom of the page so that you know the format.
• 6-2-21: Today we finished material from chapter 4 and introduced some ideas from chapter 5. We will touch on what we can of chapters 5 and 6 on Monday. Tomorrow is Exam I.
• 6-1-21: We showed, using the two step subring test, that the image of a ring homomorphism is a subring. We looked at divisibility and polynomials over rings, in particular we discussed the Division Algorithm for polynomials and proved the Remainder and Factor Theorems. Finally we defined what it means for a polynomial to be irreducible/reducible over a field. I am adding the terms domain, codomain, and range/image to your vocabulary for exam 1.
• 5-27-21: Today we covered material from section 3.3 on homomorphisms, injective, surjective, and isomorphisms. We also discussed domain, codomain, and range (image) which should be review.
• 5-26-21: Today we covered material from section 3.2 focussed on the basic properties of rings and fields.
• 5-25-21: Today we discussed Rings, Integral Domains, and Fields. Between yesterday and today we have covered the material you should need to look at sections 1.3, 2.3, and 3.1 in the text. We will be covering 3.2 in tomorrow’s class.
• 5-24-21: Reviewed Number Theory and Proofs (slides) and looked at an overview of some Basic Definitions (slides) for the course.

## Exam Guides:

### Unit 1:

• Practice: Look at the A-Type Exercises in sections 1.3, 2.3, 3.1, 3.2, & 3.3.
• Definitions: Ring, Field, Integral Domain, Homomorphism, Isomorphism, Equivalence Relation, Equivalence Class, Reflexive, Symmetric, Transitive, Modular Equivalence, Divisible, Surjective (onto), Injective(1-1), Domain of a function, Codomain of a function, Range/Image of a function.
• Theorems:
• Bezout’s Theorem: If a and b are integers, then there exist integers m and n such that ma+nb=(a,b).
• Theorem: If R is a finite ring, then every non-zero element of R is either a zero divisor or unit.
• Theorems 3.3, 3.4,  and 3.5: Basic Properties of Rings
• Theorem 3.6: on Subrings
• Theorem 3.9: All finite integral domains are fields.
• Theorem 3.10 and Corollary 3.11: Properties of Homomorphisms and Isomorphisms

### Unit 2:

• Practice: Look at the A-Type Exercises in sections 4.1-4.5, 5.1-5.3, 6.1-6.2
• Definitions: Divisibility for Polynomials, Polynomial Rings, Quotient Structures, Polynomial, Kernel, Reducible Polynomial over a Field, Irreducible Polynomial over a Field, Equivalence Relation, Equivalence Class, Ideal in a Ring, …
• Theorems:
• Theorem 4.1 (p.86): Basic Properties of $$R[x]$$
• Theorem 4.2/Corollary 4.4 (p.89): $$deg(f\cdot g)\leq deg(f) + deg(g)$$ with equality if the ring is an integral domain.
• Theorem 4.6 (p.91): Division Algorithm
• Theorem 4.15 (p. 107): Remainder Theorem
• Theorem 4.16 (p.107): Factor Theorem
• Theorem 4.24 (p.116): Eisenstein’s Criterion
• Theorem 4.26 (p.120): Fundamental Theorem of Algebra
• Theorem 5.10 (p.135): A polynomial $$p(x)\in F[x]$$ is irreducible if and only if $$F[x]/p(x)$$ is a field.
• Theorem 6.13 (p.157): First Isomorphism Theorem

### Unit 3:

• Practice: Look at the A-Type Exercises in sections 7.1-7.4 and 8.1-8.4
• Definitions: Group, Subgroup, Normal Subgroups, Cosets, Equivalence Relation, Equivalence Classes, Congruence Modulo a Subgroup, Quotient Groups, Order of a Group, Index of a Subgroup, Group Homomorphism, Group Isomorphism, Kernel of a Homomorphism, Symmetric Group, Dihedral Group, Order of an Element, Cyclic Group, Abelian Group, Center of a Group (Cayley’s Theorem???)
• Theorems:
• Theorem 7.5 (and corollary 7.6, p. 196): On Identity, Inverses, and Cancellation
• Theorem 7.9 (p. 200): On Orders
• Theorem 7.11 & 7.12 (p.204-205): Identifying Subgroups
• Theorem 7.19 (p.219): Cyclic Groups
• Theorem 7.21 (p.221): Cayley’s Theorem,
• Theorem 8.5 (p.241): Lagrange’s Theorem,
• Theorem 8.7 (p.242): Groups of Prime Order
• Theorem 8.13 (p.255): Quotient Groups
• Theorem 8.16 (p.264): The Kernel is a Normal Subgroup
• Theorem 8.20 (p.266): First Isomorphism Theorem (for groups)
• (Theorem 9.14 (p.299): Cauchy’s Theorem)

### Unit 4:

• Practice: Look at the A-Type Exercises in sections 9.1-9.3
• Definitions: Abelian Groups, p-group, Direct Products, Equivalence Relation, Equivalence Classes, Cyclic Group, Sylow p-Subgroup, Centralizer of an Element, Conjugate (conjugation)
• Theorems:
• Theorems 9.1 & 9.3 (p.283-285): Deconstructing Groups
• Lemma 9.4/Theorem 9.5 (p.290-291): Deconstructing Abelian Groups In Particular
• Theorem 9.7 (p.293): Fundamental Theorem of Abelian Groups,
• Theorem 9.13 (p.299): Sylow’s Theorem,
• Theorem 9.14 (p.299): Cauchy’s Theorem