\n\n\n\n
Class Schedule Changes:<\/h1>\n\n\n\n
Due to snow days and cancellations there will need to be a number of changes to the rest of the semester.The most significant is that unit 3 and 4 exams have been combined into a single unit exam on Group Theory. The exam for this combined unit will be on 5\/11\/2026 during final exam week. You still need to complete separate assignments for the material that would have been in unit 3 and unit 4, the unit 3 work should be handed in on Monday 4\/29\/2026 and the unit 4 content is due by 3pm on Friday 5\/15\/2026. The exam for unit 2 has been moved to Monday 4\/13\/2026. The unit 2 assignment is now due on Monday 4\/20\/2025. Finally, below on the calendar I have mapped out which text book sections should be covered and when.<\/p>\n\n\n\n
\n\n\n\n
Calendar:<\/h1>\n\n\n\n
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. <\/p>\n\n\n\n
- \n
- …<\/li>\n\n\n\n
- 5\/11\/2026:<\/em> Unit 3\/4 Exam<\/em><\/li>\n\n\n\n
- 5\/6\/2026:<\/em> Cauchy’s Theorem<\/em><\/li>\n\n\n\n
- 5\/4\/2026:<\/em> First Isomorphism Theorem for Groups<\/em><\/li>\n\n\n\n
- 4\/29\/2026:<\/em> Lagrange’s Theorem<\/em><\/li>\n\n\n\n
- 4\/27\/2026:<\/em> Cayley’s Theorem<\/em><\/li>\n\n\n\n
- 4\/22\/2026:<\/em> Today we went over a bunch of examples of groups, in particular if \\(R\\) is a ring then \\((R,+)\\) and \\((R^*,\\times)\\) are always groups. We also discussed the dihedral groups<\/em><\/strong>, \\(D_n\\) of rigid motions of an \\(n\\)-gon and symmetric groups<\/em><\/strong>, \\(S_n\\), of permutations of \\(n\\) objects. We looked at properties of groups, many were very familiar since they are the analogous to ring properties. In particular we used the two-step subgroup test<\/em><\/strong> to show that the set of matrices of the form \\[M=\\begin{pmatrix} 1 & a \\\\ 0 & b \\end{pmatrix}\\] form a subgroup, using multiplication, of the group of all invertible \\(2\\times 2\\) matrices. We also discussed orders of groups<\/em><\/strong> and orders of elements<\/em><\/strong>. Finally, we looked at a Cayley Table<\/em><\/strong> for the group \\(D_3\\) and noted that in each row we got a permutation of all the elements of the group; this leads us into the proof of Cayley’s Theorem<\/em><\/strong> which we will discuss next week. You should now look through material in sections 7.1, 7.2, and 7.3 in the text.<\/li>\n\n\n\n
- 4\/20\/2026: <\/em>We mostly finished up the Groups and Symmetries Packet<\/a> in class, finish that up at home and try to get it in some time next week. <\/li>\n\n\n\n
- 4\/15\/2026: <\/em>Unit 2 Exam<\/li>\n\n\n\n
- 4\/13\/2026:<\/em> We started going through the Groups and Symmetries Packet<\/a>. We finished the material on the triangles and will look at the square and the general permutations sections on Monday 4\/20\/2026<\/li>\n\n\n\n
- 4\/8\/2026:<\/em> Today we looked at a general proof of the Chinese Remainder Theorem for Integers (Section 14.1), htne briefly discussed how it might be applied (Section 14.2) and finally looked at the general statement of the Chinese Remainder Theorem for Rings (Section 14.3) and how it relates to the specific case when \\(R=\\mathbb{Z}\\). We also did an example of Lagrange Interpolation and how it is using the Chinese Remainder Theorem when \\(R=\\mathbb{Q}[x]\\). The Exam for Unit 2 is moved to next wednesday; monday we will spend a little time answering questions and then start the unit on groups.<\/li>\n\n\n\n
- 4\/6\/2026:<\/em> We looked at an additional example of the First Isomorphism Theorem and then at its proof. Then we connected it to the previous big result regarding quotients since if we can use them to impose an arithmetic on other sets. For example we can define addition, subtraction, multiplication, and multiplicative inverses for bytes of 1s and 0s by creating an isomorphism from \\(\\mathbb{Z}_2[x]\/(p(x))\\) to the set of bytes, where \\(p(x)\\) is an irreducible polynomial over \\(\\mathbb{Z}_2\\). We then did an example of solving system of modular equations using the Chinese Remainder Theorem. So, at this point we have finished the basics from chapters 5 and 6 that we are going to cover and done an example of the work in section 14.1. Kenneth should be typing up definitions\/theorems\/notes for today.<\/li>\n\n\n\n
- 4\/1\/2026:<\/em> As of the end of class today we have covered material from sections 5.1, 5.2, 5.3, 6.1, and much of 6.2. We will finish this up next time and likely move on to chapter 14 to keep things moving. Please remind me next time to discuss how we define when two cosets are equal” “\\(r+I=s+I\\) if and only if \\(r-s\\in I\\).” I had bypassed this because of our focus on particular examples, but we should consider it. Finally, for the notes we said Andrew should do last Wednesday’s notes, Matt should do Monday’s, and Brianna should do todays.<\/li>\n\n\n\n
- 3\/30\/2025:<\/em> We covered the background material in order to understand the theorem: If \\(F\\) is a field then \\(p(x)\\in F[x]\\) is an irreducible polynomial if and only if \\(F[x]\/(p(x))\\) is a field. In particular we reviewed what it means for a polynomial over a field to be irreducible, we learned about ideals in rings, cosets, and arithmetic in quotient rings. For the Equivalence Relations Packet<\/a><\/em>, you should finish up the sections on integer modular arithmetic and on polynomial modular arithmetic, you can also do the last section for extra credit, try to get this turned in by the 15th of April<\/strong><\/em>.<\/li>\n\n\n\n
- 3\/25\/2026:<\/em> We continued working through the Equivalence Relations Packet<\/a><\/em>, you should try to finish up\/look over the rest of section 3; we will touch on section 4, briefly, next time before moving on to chapter 5. Make sure you are comfortable with the ideas of Congruence Relations and Classes, these are key foundations for chapter 5. Finally note that since we finished the basic material from 4.1-4.4 last class you should be looking over the Type-A questions from these sections to make sure you have a handle on how some of the ideas we covered are applied; these are the questions that will eventually appear on part 2 of the exam.<\/li>\n\n\n\n
- 3\/23\/2026: Today we covered the foundational content from Sections 4.1-4.4 regarding polynomials. We also discussed the first couple of pages of the Equivalence Relations Packet<\/a><\/em>.<\/li>\n\n\n\n
- 3\/11\/2026: Class cancelled. I will post some notes and calendar updates soon.<\/li>\n\n\n\n
- 3\/9\/2026: Today we went over some problems from the exam emphasizing understanding and using definitions when answering questions. You then got the exams back; redo questions are due the Monday after break and like all out of class work must be type up neatly and professionally. The new material we covered was on polynomials. Specifically we looked at products and quotients, general formulas for coef. in a product, degrees of polynomials in rings vs integral domains, the division algorithm for polynomials over a field (the proof of which uses induction on the degree of the dividend), and briefly the Euclidean Alg. for Polynomials (I owe you a clean example of this). We will continue discussing polynomials on Wednesday. <\/li>\n\n\n\n
- 3\/4\/2026: Exam 1 (finally)<\/li>\n\n\n\n
- 3\/2\/2026: We went over the mathematica behind Chapter 13 and and RSA encryption algorithm. In particular we played with Fermat’s Little Theorem, Euler’s \\(\\phi\\)-function, and Euler’s Theorem. The first exam is on Wednesday, be sure to study the vocabulary and A-Type questions form all the material we have covered: Chapters 1, 2, 3, & 13. I have updated the vocabulary below in the exam study section.<\/li>\n\n\n\n
- 2\/25\/2026:
![\"picture](\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2024\/02\/snow_day-e1769439819504.png\")
Snow Day With, yet another, the snow day we will need to shift the exam, again, so we will discuss Chapter 13 on Monday and have the exam on 3\/4\/2026.<\/li>\n\n\n\n- 2\/23\/2026:
Snow Day With the snow day we will need to shift the exam, again, so we will discuss Chapter 13 on Wednesday and have the exam on 3\/2\/2026.<\/s><\/li>\n\n\n\n- 2\/18\/2026: We discussed properties of homomorphisms, looked at some additional examples of homomorphisms, and some subtleties in some of those examples.<\/li>\n\n\n\n
- 2\/11\/2025: Today we introduced homomorphisms and isomorphisms. Particular examples included conjugation, the identity map, the zero map, and complex conjugation. We will pick up next wednesday discussing 1-1 and onto. We are moving the exam for unit 1 to 2\/25\/2026; try to get the assignment in by March 2nd. <\/li>\n\n\n\n
- 2\/9\/2026: We finished covering ideas from 3.1 and 3.2. We didn’t cover every single tiny detail, but we covered a broad overview so that you can read the sections and follow along. We will cover section 3.3 next time. Also, I will review material for unit one and either shift the schedule moving the exam to Wednesday the 25th or we will cut out some material. I will give a decision Wednesday based on how class goes. P.S. Please recall that Exam Guides are listed below<\/em><\/strong>.<\/li>\n\n\n\n
- 2\/4\/2026: We finished overing some background number theory from chapters 1 and 2. We then started looking at properties of Rings from sections 3.1 and 3.2. We will pick up there on Monday.<\/li>\n\n\n\n
- 2\/2\/2026: Today we covered some of the key number theory material from chapters 1 and 2. In particular we discussed: Divisibility, the Well Ordering Principle, Bezout’s Lemma, the Fundamental Theorem of Arithmetic, and various lemmas such as “Lemma: \\(p\\in\\mathbb{Z}\\) is prime if and only if whenever \\(p|ab\\) we know \\(p|a\\) or \\(p|b\\).” We also did examples of direct, contrapositive, contradiction, and biconditional proofs. Finally, we discussed the importance of paying attention to the details of definition and theorem statements in particular paying attention to words like least, greatest, and unique.<\/li>\n\n\n\n
- 1\/28\/2026: Today we went through most the rest of the Algebraic Structures Packet (https:\/\/tinyurl.com\/2s36znfb)<\/a>. 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.<\/li>\n\n\n\n
- 1\/26\/2026:
Snow Day : 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)<\/a>, and then try to complete the sections in the intro packet on integers modulo 6 and integers modulo 7.<\/li>\n\n\n\n- 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.<\/li>\n<\/ul>\n\n\n\n
\n\n\n\nAssignments:<\/h1>\n\n\n\n
For each unit select 5 problems from different sections from the exercises
listed below.<\/p>\n\n\n\nFlashcards and Notes:<\/h2>\n\n\n\n
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.<\/p>\n\n\n\n
- \n
- Sammi Silvestri<\/li>\n\n\n\n
- Andrew Tagg<\/li>\n\n\n\n
- Lorelei Stancavage<\/li>\n\n\n\n
- Matthew Purr<\/li>\n\n\n\n
- Brianna Hernandez<\/li>\n\n\n\n
- Kenneth Scholl<\/li>\n\n\n\n
- Teddy Blanchard<\/li>\n\n\n\n
- Jennifer Altamirano<\/li>\n\n\n\n
- Repeat<\/li>\n<\/ol>\n\n\n\n
Unit 1<\/h2>\n\n\n\n
- \n
- Section 1.3: #’s 24, 34<\/li>\n\n\n\n
- Section 2.3: #14<\/li>\n\n\n\n
- Section 3.1: #’s 22, 32, 42<\/li>\n\n\n\n
- Section 3.2: #’s 28, 32<\/li>\n\n\n\n
- Section 3.3: #’s 16, 24, 30, 36<\/li>\n\n\n\n
- Chapter 13: #’s 3, 5, 6<\/li>\n<\/ul>\n\n\n\n
Unit 2: Due Monday 4\/20\/2026<\/h2>\n\n\n\n
- \n
- Section 4.1: #’s (11 & 15) as one problem<\/li>\n\n\n\n
- Section 4.2: #8<\/li>\n\n\n\n
- Section 4.3: #16<\/li>\n\n\n\n
- Section 4.4: #14<\/li>\n\n\n\n
- Section 5.1: #8<\/li>\n\n\n\n
- Section 5.3: #2<\/li>\n\n\n\n
- Section 6.1: #’s(6 & 24) as one problem<\/li>\n\n\n\n
- Section 14.1: #’s 14, 18, 20<\/li>\n\n\n\n
- Section 14.2: #’s 6, 7<\/li>\n\n\n\n
- Section 14.3: #’s 4, 5, 6<\/li>\n<\/ul>\n\n\n\n
Unit 3: Wednesday Due 4\/29\/2026<\/h2>\n\n\n\n
- \n
- Section 7.1: #’s 20, 24, 26, 28<\/li>\n\n\n\n
- Section 7.2: #’s 20, 25, 26, 30 (use #28)<\/li>\n\n\n\n
- Section 7.3: #’s 19, 39, 42, 48 (hint: If \\(\\mathbb{R}^*=<a>\\), then for some \\(k\\in\\mathbb{Z}\\), \\(1=a^k\\).)<\/li>\n\n\n\n
- Section 7.4: #\u2019s 26, (28 & 29) as a single problem, 30, 32,36<\/li>\n\n\n\n
- Section 7.5: #’s 22, 24, 30 (use #29)<\/li>\n<\/ul>\n\n\n\n
Unit 4: Due 5\/15\/2026 by 3pm<\/h2>\n\n\n\n
- \n
- Section 8.1: #’s 24, 26, 29<\/li>\n\n\n\n
- Section 8.2: #’s 16, 18, 20, 26<\/li>\n\n\n\n
- Section 8.3: #’s 16 (look at the solution to #15), 22, 24, 32<\/li>\n\n\n\n
- Section 8.4: #’s22, 24, 26<\/li>\n\n\n\n
- Section 16.1 #’s (12&13) as one, 18<\/li>\n\n\n\n
- Section 16.2: #’s 12, 14<\/li>\n<\/ul>\n\n\n\n
Comments & Advice on Homework:<\/h2>\n\n\n\n
The following are comments that I have made in previous semesters which I believe are worth mentioning here.<\/p>\n\n\n\n
- \n
- Take time to glance at other problems.<\/strong><\/em> 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.<\/li>\n\n\n\n
- Use previous results we have discussed, either as review or new material, in class.<\/strong><\/em> 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.):\n
- \n
- If \\(c|ab\\) and \\((c,a)=1\\), then \\(c|b\\).<\/li>\n\n\n\n
- If \\(p|ab\\) and \\(p\\) is prime, then \\(p|a\\) or \\(p|b\\).<\/li>\n\n\n\n
- 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\\).<\/li>\n\n\n\n
- 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.<\/li>\n\n\n\n
- If \\(f:R\\rightarrow S\\) is a homomorphism, Theorem 3.10 tells us things like:\n
- \n
- \\(f(-a)=-f(a)\\),<\/li>\n\n\n\n
- \\(f(a^{-1})=f(a)^{-1}\\),<\/li>\n\n\n\n
- \\(f(0)=0\\), and <\/li>\n\n\n\n
- \\(f(1)=1\\).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n
- Using proper terminology and notation is faster and clearer.<\/strong><\/em> It is easier to say “we will show the operation is commutative” than it is to type out what that means all the time.<\/li>\n\n\n\n
- Try to plan out your work before you start typing.<\/strong><\/em> 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.<\/li>\n\n\n\n
- You are welcome to ask for hints or to ask questions, but be specific.<\/strong><\/em> 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.<\/li>\n\n\n\n
- Be sure you are doing your own homework.<\/strong><\/em> 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. <\/li>\n<\/ol>\n\n\n\n
\n\n\n\nExam Guides:<\/h1>\n\n\n\n
Unit 1:<\/strong><\/h2>\n\n\n\n
- \n
- Practice:<\/strong><\/span><\/em> 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)<\/li>\n\n\n\n
- Definitions:<\/strong><\/em><\/span> Ring, Field, Integral Domain, Homomorphism, Isomorphism, Equivalence Relation, Equivalence Class, Reflexive, Symmetric, Transitive, Modular Equivalence, Divisible, Surjective (onto), Injective(1-1), Inverses, Units, Unity, Zero Divisors, Euler’s \\(\\phi\\)-Function<\/li>\n\n\n\n
- Theorems:<\/strong><\/span><\/em>\n
- \n
- Bezout’s Theorem:<\/strong> If <\/em>a and <\/em>b are integers, then there exist integers <\/em>m and <\/em>n such that <\/em>ma+nb=(a,b).<\/em><\/li>\n\n\n\n
- Theorem:<\/strong> If <\/em>R is a finite ring, then every non-zero element of <\/em>R is either a zero divisor or unit.<\/em><\/li>\n\n\n\n
- Theorems 3.3, 3.4, and 3.5: <\/strong>Basic Properties of Rings<\/li>\n\n\n\n
- Theorem 3.6:<\/strong> on Subrings<\/li>\n\n\n\n
- Theorem 3.9:<\/strong> All finite integral domains are fields.<\/li>\n\n\n\n
- Theorem 3.10 and Corollary 3.11:<\/strong> Properties of Homomorphisms and Isomorphisms<\/li>\n\n\n\n
- Lemma 13.2 Fermat’s Little Theorem<\/strong><\/li>\n\n\n\n
- …<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
Unit 2:<\/strong><\/h2>\n\n\n\n
- \n
- Practice:<\/strong><\/span><\/em> 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)<\/li>\n\n\n\n
- Definitions:<\/strong><\/em><\/span> 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, …<\/li>\n\n\n\n
- Theorems:<\/strong><\/span><\/em>\n
- \n
- Theorem 4.1 (p.86):<\/strong> Basic Properties of \\(R[x]\\) <\/li>\n\n\n\n
- Theorem 4.2\/Corollary 4.4 (p.89):<\/strong> \\(deg(f\\cdot g)\\leq deg(f) + deg(g)\\) with equality if the ring is an integral domain.<\/li>\n\n\n\n
- Theorem 4.6 (p.91):<\/strong> Division Algorithm<\/li>\n\n\n\n
- Theorem 4.15 (p. 107):<\/strong> Remainder Theorem<\/li>\n\n\n\n
- Theorem 4.16 (p.107):<\/strong> Factor Theorem<\/li>\n\n\n\n
Theorem 4.24 (p.116):<\/strong> Eisenstein’s Criterion<\/s><\/li>\n\n\n\n- Theorem 4.26 (p.120):<\/strong> Fundamental Theorem of Algebra<\/li>\n\n\n\n
- Theorem 5.10 (p.135):<\/strong> A polynomial \\(p(x)\\in F[x]\\) is irreducible if and only if \\(F[x]\/p(x)\\) is a field.<\/li>\n\n\n\n
- Theorem 6.13 (p.157):<\/strong> First Isomorphism Theorem <\/li>\n\n\n\n
- Theorems 14.? and 14.?:<\/strong> Chinese Remainder Theorems<\/li>\n\n\n\n
- …<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
Unit 3:<\/strong><\/h2>\n\n\n\n
- \n
- Groups and Subgroups Slides v0.3<\/em><\/strong><\/a><\/li>\n\n\n\n
- Groups and Homomorphism Slides v0.2<\/em><\/strong><\/a><\/li>\n\n\n\n
- Practice:<\/strong><\/span><\/em> 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)<\/li>\n\n\n\n
- Definitions:<\/strong><\/em><\/span> 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<\/li>\n\n\n\n
- Theorems:<\/strong><\/em><\/span>\n
- \n
- Theorem 7.5 (and corollary 7.6, p. 196):<\/strong> On Identity, Inverses, and Cancellation<\/li>\n\n\n\n
- Theorem 7.9 (p. 200):<\/strong> On Orders<\/li>\n\n\n\n
- Theorem 7.11 & 7.12 (p.204-205):<\/strong> Identifying Subgroups<\/li>\n\n\n\n
- Theorem 7.19 (p.219):<\/strong> Cyclic Groups<\/li>\n\n\n\n
- Theorem 7.21 (p.221):<\/strong> Cayley’s Theorem<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
Unit 4:<\/strong><\/h2>\n\n\n\n
- \n
- Cosets and Lagrange’s Theorem<\/em><\/strong><\/a><\/li>\n\n\n\n
- Quotients and Isomorphisms v0.4<\/a><\/strong><\/em><\/li>\n\n\n\n
- Practice:<\/strong><\/em><\/span> 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)<\/li>\n\n\n\n
- Definitions:<\/strong><\/em><\/span> 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, …<\/li>\n\n\n\n
- Theorems:<\/strong><\/span>\n
- \n
- Theorem 8.5 (p.241):<\/strong> Lagrange’s Theorem,<\/li>\n\n\n\n
- Theorem 8.7 (p.242):<\/strong> Groups of Prime Order<\/li>\n\n\n\n
- Theorem 8.13 (p.255):<\/strong> Quotient Groups<\/li>\n\n\n\n
- Theorem 8.16 (p.264):<\/strong> The Kernel is a Normal Subgroup<\/li>\n\n\n\n
- Theorem 8.20 (p.266):<\/strong> First Isomorphism Theorem (for groups)<\/li>\n\n\n\n
- …<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
- Theorem 8.7 (p.242):<\/strong> Groups of Prime Order<\/li>\n\n\n\n
- Quotients and Isomorphisms v0.4<\/a><\/strong><\/em><\/li>\n\n\n\n
- Theorem 7.9 (p. 200):<\/strong> On Orders<\/li>\n\n\n\n
- Groups and Homomorphism Slides v0.2<\/em><\/strong><\/a><\/li>\n\n\n\n
- Theorem 4.2\/Corollary 4.4 (p.89):<\/strong> \\(deg(f\\cdot g)\\leq deg(f) + deg(g)\\) with equality if the ring is an integral domain.<\/li>\n\n\n\n
- Definitions:<\/strong><\/em><\/span> 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, …<\/li>\n\n\n\n
- Theorem:<\/strong> If <\/em>R is a finite ring, then every non-zero element of <\/em>R is either a zero divisor or unit.<\/em><\/li>\n\n\n\n
- Definitions:<\/strong><\/em><\/span> Ring, Field, Integral Domain, Homomorphism, Isomorphism, Equivalence Relation, Equivalence Class, Reflexive, Symmetric, Transitive, Modular Equivalence, Divisible, Surjective (onto), Injective(1-1), Inverses, Units, Unity, Zero Divisors, Euler’s \\(\\phi\\)-Function<\/li>\n\n\n\n
- Try to plan out your work before you start typing.<\/strong><\/em> 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.<\/li>\n\n\n\n
- Use previous results we have discussed, either as review or new material, in class.<\/strong><\/em> 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.):\n
- Take time to glance at other problems.<\/strong><\/em> 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.<\/li>\n\n\n\n
- 5\/6\/2026:<\/em> Cauchy’s Theorem<\/em><\/li>\n\n\n\n