Algebraic Structures

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.

  • …
  • 1/28/2026: Today we went through most the rest of the Algebraic Structures Packet (https://tinyurl.com/2s36znfb). Try and finish this up, which you may do in writing, by next Wednesday. We will cover key material from chapters 1 and 2 on Monday.
  • 1/26/2026: picture of a snowing cloud Snow Day picture of a snowing cloud: For Wednesday make sure that you have finished the material on polynomials, look at the first couple of sections of this slide deck: Modular Arithmetic (https://tinyurl.com/z63nrt45), and then try to complete the sections in the intro packet on integers modulo 6 and integers modulo 7.
  • 1/21/2026: Today we started working through the introduction to algebraic structures packet. We finished the sections on integers, \(\mathbb{Z}\), rationals, \(\mathbb{Q}\), and complex numbers \(\mathbb{Z}\). You needed to try and finish the section on polynomials before next class.

Assignments:

For each unit select 5 problems from different sections from the exercises
listed below.

Flashcards and Notes:

Remember that as part of your in class grade you need to take turns taking notes. The URL to edit the Flashcards and Notes is on Blackboard since it wouldn’t be safe to make those public. The order in which you need to take the notes is listed below.

  1. Sammi Silvestri
  2. Andrew Tagg
  3. Lorelei Stancavage
  4. Matthew Purr
  5. Brianna Hernandez
  6. Kenneth Scholl
  7. Teddy Blanchard
  8. Jennifer Altamirano
  9. Repeat

Unit 1

  • 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
  • Chapter 13: #’s 3, 5, 6

Unit 2

  • 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
  • Section 14.1: #’s 14, 18, 20
  • Section 14.2: #’s 6, 7
  • Section 14.3: #’s 4, 5, 6

Unit 3

  • Section 7.1: #’s 20, 24, 26, 28
  • Section 7.2: #’s 20, 25, 26, 30 (use #28)
  • Section 7.3: #’s 19, 39, 42, 48 (hint: If \(\mathbb{R}^*=<a>\), then for some \(k\in\mathbb{Z}\), \(1=a^k\).)
  • Section 7.4: #’s 26, (28 & 29) as a single problem, 30, 32,36
  • Section 7.5: #’s 22, 24, 30 (use #29)

Unit 4

  • Section 7.4: #’s 26, (28 & 29) as a single problem, 30, 32
  • Section 8.1: #’s 24, 26, 29
  • Section 8.2: #’s 16, 18, 20, 26
  • Section 8.3: #’s 16 (look at the solution to #15), 22, 24, 32
  • Section 8.4: #’s22, 24, 26
  • Section 16.1 #’s (12&13) as one, 18
  • Section 16.2: #’s 12, 14

Comments & Advice on Homework:

The following are comments that I have made in previous semesters which I believe are worth mentioning here.

  1. Take time to glance at other problems.  For example, many of you tried Section 1.3 #24, if you had glanced at #23 you would see that it is very similar and tells you to use the result in #19.  The solutions to both of these are in the back of the text.
  2. Use previous results we have discussed, either as review or new material, in class.  All of the following are things we have touched on that would be useful for some of the problems (This is not an exhaustive list, it is just a few examples.):
    • If \(c|ab\) and \((c,a)=1\), then \(c|b\).
    • If \(p|ab\) and \(p\) is prime, then \(p|a\) or \(p|b\).
    • Given \(a,b,c\in\mathbb{Z}\), \(c|a\) and \(c|b\) if and only if \(c\) divides all linear combinations of \(a\) and \(b\).
    • If \(a\in R\) is a unit, then it is not a zero divisor.  Likewise, if \(a\in R\) is a zero divisor then it is not a unit.
    • If \(f:R\rightarrow S\) is a homomorphism, Theorem 3.10 tells us things like:
      • \(f(-a)=-f(a)\),
      • \(f(a^{-1})=f(a)^{-1}\),
      • \(f(0)=0\), and 
      • \(f(1)=1\).
  3. Using proper terminology and notation is faster and clearer.  It is easier to say “we will show the operation is commutative” than it is to type out what that means all the time.
  4. Try to plan out your work before you start typing.  For example in Section 3.1 #22, if you prove early on that \(a\oplus b=b\oplus a\) and \(a\odot b=b\odot a\), then some other properties are easier to prove.  You don’t have to be a slave to the order the properties are listed in.
  5. You are welcome to ask for hints or to ask questions, but be specific.  I don’t mind trying to help you out if you have specific questions, but I won’t solve the problem for you and I won’t pre-grade all your work.  You should make an effort to ask very specific questions; it is helpful if you try to write them down ahead of time.
  6. Be sure you are doing your own homework.  Some of you turned in work that was very similar to one another and similar to some solutions that can be found online.  I have been very clear in class that this sort of behavior is absolutely unacceptable.  I didn’t hand out any zeros this time, but I wanted to be clear that I do notice and this should not continue. 

Exam Guides:

Unit 1:

  • Practice: Look at the A-Type Exercises in sections 1.3, 2.3, 3.1, 3.2, 3.3, and chapter 13. (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, & chapter 14. (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:

  • Groups and Subgroups Slides v0.3
  • Groups and Homomorphism Slides v0.2
  • 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:

  • Cosets and Lagrange’s Theorem
  • Quotients and Isomorphisms v0.4
  • Practice: Look at the A-Type Exercises in sections 8.1-8.4, 16.1, & 16.2. (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: