Algebraic Structures Spring 2023

Syllabus

Spring 2023 Final Exam Schedule:

  • Monday 5/8: Office Hours 11-12 and 1-2, MAT 375 Exam 2-4:30pm
  • Tuesday 5/9: Office Hours 11-12 and 1-2, MAT 453 Final Talks 2-4:30pm
  • Thursday 5/11: MAT 170 Exam 8-10:30am, Office Hours 11-12

Comment on Homework:

I have finished grading assignment 1, from those who handed it in, and have a few comments.

  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.):
    1. If \(c|ab\) and \((c,a)=1\), then \(c|b\).
    2. If \(p|ab\) and \(p\) is prime, then \(p|a\) or \(p|b\).
    3. Given \(a,b,c\in\mathbb{Z}\), \(c|a\) and \(c|b\) if and only if \(c\) divides all linear combinations of \(a\) and \(b\).
    4. 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.
    5. 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. 

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.

  • 4/24 – Today we finished off new material for the course. Thursday is review. There will be a make-up quiz and after the quiz we will answer questions if your bring them. The updated version of today’s slides have been posted. You should probably read through the slides, they have a lot of important information.
  • 4/20 – Today we continued our discussion of normal subgroups. In particular we saw that if \(N\subseteq G\) is a normal subgroup then the set of all left cosets \(G/N=\left\{gN\middle |g\in G\right\}\) is also a group; this means that every normal subgroup is the kernel of some homomorphism. We also saw that every kernel is a normal subgroup. We will formalize this connection between \(G/N\) and homomorphisms in the next class when we examine the First Isomorphism Theorem for Group Homomorphisms.
  • 4/17 – Today we finished the slides on Cosets and Lagrange’s Theorem and started the slides on Quotients and Isomorphisms. So far we have only looked at some properties of Normal Subgroups. Specifically we saw that \(N\subseteq G\) is a normal subgroup if and only if \(\forall g\in G: N=gNg^{-1}\). Using this we saw that \(\left<r^2\right>\subseteq D_4\) is a normal subgroup and \(\left<f\right>\subseteq D_4\) is not. In the next class we will look at the connection between kernels of homomorphisms and normal subgroups.
  • 4/13 – Exam 3
  • 4/10 – We completed the proof of Cayley’s Theorem, the last item in Unit 3. We then discussed cosets of subgroups which is the first topic in Unit 4. On Thursday is your exam on Unit 3, be sure to look at the study guide below, and next Monday we will pick up our discussion of cosets by looking at some of their properties.
  • 4/6 – We went over the slides on homomorphisms a second time as well as looking at the material on isomorphisms and group actions. We will review the group actions materials next class and look at Cayley’s Theorem. The Groups and Homomorphism Slides v0.2 have been updated to include material on centers and the centralizer. Remember, as was announced in class and posted previously, the Unit 3 exam has been moved to 4/13.
  • 4/3 – We covered symmetric groups and started discussing group homomorphisms. To give us time to adequately cover material we will move the Unit 3 exam to 4/13. We will start covering unit 4 material on 4/10.
  • 3/30 – Today we looked at some additional examples of groups and subgroups. Our list of groups now includes \((\mathbb{Z},+),(\mathbb{Z}_n,+), (U_n,\times), (\mathbb{R},+),(\mathbb{R}^*,\times), (D_n,\circ)\), and \((S_n,\circ)\). We looked at some basic properties of groups and subgroups and discussed cyclic groups. We ended class by looking at numerous examples of using cycle notation, we will pick up there on Monday.
  • 3/27 – Starting today we are doing make-up quizzes each day we don’t have an exam. Information on the make-up quizzes is here: Make-Up Quiz Guide. You are responsible for keeping track of material covered in class. Today after the quiz we reviewed the material on groups that we discussed last Monday, looked at a theorem on orders of group elements, and learned some new vocabulary while looking at the examples of \(D_n\), \(\mathbb{Z}\), and \(\mathbb{Z}_n\), and \(\mathbb{Z}\oplus\mathbb{Z}_n\). We will start looking at Symmetric Groups next class. P.S. the typos in the slides have been fixed: Groups and Subgroups Slides v0.3.
  • 3/23 – Exam 2 will be on this day
  • 3/20 – Today we introduced groups by looking at dihedral groups, \(D_n\), which are the transformations of regular \(n\)-sided polygons and at the group \(\mathbb{Z}\oplus\mathbb{Z}_2\) which is the combination of two separate groups. The slides for this unit are posted below with the Unit 3 Exam Guide. We will continue this discussion next Monday.
  • 3/9/2023 – We introduced the idea of kernels of a homomorphism (which is analogous to the null space of a linear transformation) and used it to prove Theorem 6.13 (p.157): First Isomorphism Theorem. Along the way we introduced ideals of a ring with which we could generalize the idea of modular equivalence to say two elements \(a,b\in R\) in a ring are equivalent modulo an ideal \(I\subseteq R\) if \((a-b)\in I\). We can write the equivalence classes modulo \(I\) as \[[a]=a+I=\{a+ri|r\in R,\ and\ i\in I\}.\] Also, arithmetic is well defined for these equivalence classes, though multiplication requires that \(I\) is a two sided ideal, \(\forall r\in R: rI=Ir=I\) (consider \((a+i)(b+j)\) for \(i,j\in I\) and \(a,b\in R\) and decide why we need \(rI=Ir\) for multiplication to be well defined).
  • 3/6/2023 – We finished the Fundamental Theorem of Algebra and 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.
  • 3-2-2023 – Fundamental Theorem Handout
  • 2-27-2023 – We practiced using the division algorithm and finding gcd’s with polynomials in \(\mathbb{Q}[x]\) and we looked in detail at the proof of the Rational Root Theorem. We will finish off material from Chapter 4 and maybe start material in chapter 5 in the next class.
  • 2-23-2023 – Today we started by reviewing our previous discussion of polynomials over rings, \(R[x]\). In particular the ways in which they can behave like integers when the ring is a field, \(F[x]\), since then we get a division algorithm, Euclidian algorithm, greatest common divisors, and an analog of Bezout’s lemma. Also, we get the generalizations of the remainder and factor theorems from precalculus. Then started discussing reducibility and irreducibility of polynomials, i.e. whether or not they can be factored, and saw that this depended on the underlying ring. For example \(f(x)=x^4-5\) is irreducible over \(\mathbb{Z}\) or \(\mathbb{Q}\) but is reducible over \(\mathbb{R}\), and factors completely over \(\mathbb{C}\); \[f(x)=(x-\sqrt[4]{5})(x+\sqrt[4]{5})(x\sqrt[4]+i{5})(x-i\sqrt[4]{5}).\] The idea of irreducibility can also be extended to all rings, though for now we just care about polynomials. Toward the end of class we were working on showing that if a polynomial has a root then it is reducible (the contrapositive of which is that if it is irreducible then it has no roots). We got to the point where we wanted to claim \[0<deg(x-a),deg(q(x))<f(x)\] where \(f(a)=0\) and \(f(x)=q(x)(x-a)\); try and justify this claim for next class (Hint: Check our assumptions for the lemma.). We have now touched on material from sections 4.1-4.4.
  • 2-16-2023 – We looked at polynomials over rings, \(R[x]\), where the coefficients for the polynomials come from arbitrary rings. We saw that if the ring is a field, \(F[x]\), then the polynomials behave in much the same way as integers, e.g. we have a division algorithm, a concept of greatest common divisor, and an Euclidian algorithm. However, if the ring was not a field, or at least an integral domain, then we got unexpected behavior such as \[(8x^2+2x)(6x^5-3)=48x^7-24x^2+12x^6-6x\equiv -6x\pmod{12}\] so that the degree of a product of two polynomials can be less than the sum of their degrees. Toward the end of class we also discussed how the remainder theorem (and factor theorem), which you would have learned in precalculus or algebra 2, follow almost immediately from the division algorithm. You should now look through material from Section 4.1 and 4.2 in your text.
  • 2-13-2023 – Unit 1 Exam
  • 2-9-2023 – We studied isomorphisms today. A Ring Isomorphism is a is a homomorphism which is injective (1-1) and surjective (onto). We saw that the map \[\gamma:\mathbb{Z}_{12}\rightarrow\mathbb{Z}_3\times\mathbb{Z}_4\] defined by \(n\mapsto (n,n)\) is a an isomorphism while the similarly defined map \[\beta:\mathbb{Z}_{12}\rightarrow\mathbb{Z}_2\times\mathbb{Z}_6\] is not. The crucial difference being that \((3,4)=1\) while \((2,6)=2\).
  • 2-6-2023 – We studied homomorphisms and their properties. Particularly, we saw that \(\phi:\mathbb{Z}_n\rightarrow\mathbb{Z}_m\) is a homomorphisms when \(m|n\). And, for all homomorphisms, \(\phi(0)=0, \phi(-a)=-\phi(a), \phi(a-b)=\phi(a)-\phi(b)\), and for surjective homomorphisms from rings with unity/units we had \(\phi(1)=1\) and \(\phi(a^{-1})=\phi(a)^{-1}\).
  • 2-2-2023 – Today we carefully walked through the proofs that: All fields are integral domains; All finite integral domains are fields; and \(\mathbb{Z}_n\) is a field if and only if \(n\) is prime. Also, right at the end of class we discussed the definition of a Ring Homomorphism and looked at one example. You should be able to attempt all the exercises except those from section 3.3 and you can start reviewing material for the exam (look at the exam guide below).
  • 1-30-2023 – We looked at many of the properties that are true for all rings such as \(\forall a\in R: a\cdot 0=0\) and \(\forall a,b\in R: -a\cdot b=a\cdot(-b)=-(ab)\). We then started discussing zero divisors and units in rings in general, and in integral domains and fields in particular. We saw that every field is an integral domain. We will pick up here on Thursday by showing that every finite integral domain is a field and then that \(\mathbb{Z}_n\) is a field if and only if \(n\) is prime. You should start looking through the material from Section 3.2.
  • 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 1 problem from each section, you may do 1 more for extra credit. Due 4/17/2023
    • Section 7.1: #20, #24, #26, #28
    • Section 7.2: #20, #25, #26, #30 (use #28)
    • Section 7.3: #19, #39, #42, #48 (hint: If \(\mathbb{R}^*=<a>\), then for some \(k\in\mathbb{Z}\), \(1=a^k\).)
    • Section 7.5: #22, #24, #30 (use #29)
  • Unit 4: Complete four problems from different sections, you may do 1 more for extra credit. Due 5/8/2023
    • Section 7.4: #26, #’s 28 & 29 as a single problem, #30, #32
    • Section 8.1: #24, #26, #29
    • Section 8.2: #16, #18, #20, #26
    • Section 8.3: #16 (look at the solution to #15), #22, #24, #32
    • Section 8.4: #22, #24, #26
  • Extra Credit Worksheets: (+2.5% to your grade = 1/2 a missing assignment each)
    1. Complete these worksheets on algebraic structures: Algebraic Structures Intro and Overview ver 0.2
    2. Complete this packet on equivalence relations: Exploring Equivalence Relations

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:

  • 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. (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: