The calendar on the syllabus is the plan for this semester, this calendar reflects in detail what we actually get to cover in class and when.

- 1-30-2023 – …
- 1-26-2023 – We worked through this set of Example Rings (click here) to get a feel for what properties a set together with two operations, \(+\) and \(\times), can have. We concluded the discussion by labelling each as a
or**Ring, Ring with a Unit, Commutative Ring, Domain, Integral Domain, Division Ring,**. At this point we have looked at some review material and basic examples/definitions related to Rings. You should look through Chapters 1 and 2 and Section 3.1.**Field** - 1-23-2023 – We spent much of class looking at properties of arithmetic in \(\mathbb{Z}_6\), \(\mathbb{Z}_5\), \(M_2(\mathbb{R})\), \(\mathbb{Z}\), \(\mathbb{2Z}\), and \(\mathbb{Q}\). In particular we looked at what properties they had in common such as \(a+0=0+a=a\) or \(a(b+c)=ab+ac\) and properties they didn’t all share such as \(ab=ba\). We then listed off 11 distinct properties that we can look for, the first 8 of which define a
. We will continue examining rings and their properties on Thursday.**ring** - 1-19-2023 – Today we looked at the syllabus and slammed through much of this slide show Review of MAT 207: Proofs type Material. We also discussed the idea that Abstract algebra focussed on recognizing common patterns in different area/objects which allows us to common conclusions about areas of study that might not otherwise seem to have anything in common.

**All out of class work in this class must be typed up and in complete sentences.**

**Unit 1:**Complete one problem from each section and then one additional exercise for a total of 6. Due 2-23-2023- Section 1.3: #’s 24, 34
- Section 2.3: #14
- Section 3.1: #’s 22, 32, 42
- Section 3.2: #’s 28, 32
- Section 3.3: #’s 16, 24, 30, 36

**Unit 2:**Complete four of the following problems, you may then complete 2 more for extra credit. Due 3-27-2023- Section 4.1: #’s (11 & 15) as one problem
- Section 4.2: #8
- Section 4.3: #16
- Section 4.4: #14
- Section 5.1: #8
- Section 5.3: #2
- Section 6.1: #’s(6 & 24) as one problem

**Unit 3:**Complete XX of the following problems, you may do 1 more for extra credit. Due 4/17/2023- Section 7.1: #20
- Section 7.2: #25
- Section 7.3: #39
- Section 7.5: #22

**Unit 4:**Complete XX of the following problems, you may do 1 more for extra credit. Due 5/8/2023- Section 7.4: #’s 28 & 29 as a single problem
- Section 8.1: #29
- Section 8.2: #16
- Section 8.3: #24
- Section 8.4: #22

Look at the A-Type Exercises in sections 1.3, 2.3, 3.1, 3.2, & 3.3. (just look at the odd ones that have answers in the back of the book)**Practice:**Ring, Field, Integral Domain, Homomorphism, Isomorphism, Equivalence Relation, Equivalence Class, Reflexive, Symmetric, Transitive, Modular Equivalence, Divisible, Surjective (onto), Injective(1-1)**Definitions:****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

Look at the A-Type Exercises in sections 4.1-4.5, 5.1-5.3, 6.1-6.2. (just look at the odd ones that have answers in the back of the book)**Practice:**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, …**Definitions:****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

Look at the A-Type Exercises in sections 7.1-7.4. (just look at the odd ones that have answers in the back of the book)**Practice:**Group, Symmetric Group, Dihedral Group, Subgroup, Equivalence Relation, Equivalence Classes, Order of a Group, Order of an Element, Cyclic Group, Abelian Group, Group Homomorphism, Group Isomorphism, Kernel of a Homomorphism, Center of a Group**Definitions:****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,

Look at the A-Type Exercises in sections 8.1-8.4. (just look at the odd ones that have answers in the back of the book)**Practice:**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**Definitions:****Theorems:****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)

- Review of MAT 207: Proofs type Material. This slide show reviews some of the content you should recall from Proofs and examples of the types of proof techniques you would have seen. Download the file, open it in Acrobat Reader (or equivalent), change the view to full screen mode, and work through the slide show.
- “Writing Math Well” and “Guidelines for good Mathematical Writing” by Francis Su
- Writing Math Exercise
- LaTeX Quick Reference
- LaTeX Homework Template
- LaTeX Lesson Videos on media.wcsu.edu
- Group Explorer