Syllabus


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)

Links and Handouts: