## Calendar:

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 Ring, Ring with a Unit, Commutative Ring, Domain, Integral Domain, Division Ring, or Field. 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. • 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 ring. We will continue examining rings and their properties on Thursday.
• 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.

## Assignments:

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

### Exam Guides:

#### Unit 1:

• Practice: 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)
• Definitions: Ring, Field, Integral Domain, Homomorphism, Isomorphism, Equivalence Relation, Equivalence Class, Reflexive, Symmetric, Transitive, Modular Equivalence, Divisible, Surjective (onto), Injective(1-1)
• 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. (just look at the odd ones that have answers in the back of the book)
• 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. (just look at the odd ones that have answers in the back of the book)
• Definitions: 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
• 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,

#### Unit 4:

• Practice: 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)
• 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
• 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)