Section 8.5 due 12/6

I got lost in the proof of Theorem 8.30.

As I was reading this section, I kept thinking about how difficult it would be to come up with these theorems and then to go about proving them. Which comes first: the pattern to see the result of the theorem or the steps that are gone through in the theorem and then check if the idea actually works? Whatever the process is, it seems like it would be really tedious and difficult.

Section 8.3 due 12/1

I don't have specific questions, but I would like some further insight into the Second and Third Sylow Theorems because I don't feel like I understand them.

I like the First Sylow Theorem but I didn't understand it until I read the example. The notation is confusing. They should do something about that.

Section 8.2 due 11/29

I was hoping for an example to accompany more of the theorems because I don't understand them very well.

This section seemed much harder to understand than the previous sections. I just wish I could understand it better so that I could ask more questions.

Review due 11/17

I need to review a lot before taking the exam: right and left cosets, order of groups, subgropus generated by S, cyclic groups, quotient groups, etc.

I would benefit from going over 16, 24,25, or 26 of 7.7 (p. 221).

Section 7.7 due Nov 12

I don't understand the second example on p 218. How is N+(0,0)=N+(1,2) in that example?

It is interesting that cyclic quotient groups G/Z(G) imply that G is abelian. I think that is a interesting correlation.

Extra Credit Colloquim

Dr. Helen Moore talked about her career as a mathematician in the pharmaceutical field. As a researcher, she and a team of other mathematicians try to figure out what the optimal therapy for HIV and Leukemia patients are.

It was a really interesting lecture because I don't generally care about the applications that are shown in math classes--applications that are more geared toward the engineers in the classes. But this showed how math could be applied to helping fight cancer, something that we always think about medical doctors being able to achieve.

It was also interesting because of the importance of precision in the mathematics. Dr. Moore talked about doctors administering drugs to patients--different levels and different combinations of drugs. She said it was important to be accurate in their models because you wouldn't be able to give a person massive amounts of drugs without there being negative effects. (The FDA and drug companies would monitor that more, but it was still important to take into consideration in their models.)

Section 7.6 due Nov 10

Does left/right congruence affect other things? (For example, if there were a "divides" then things might get messy.) Can you further explain the definition of "normal" because they try to make a distinction at the top of p. 212 but I don't understand what they are getting at.

I was trying to decide if I like groups or rings better. I think I like groups better. You can define the group operation however you want as long as it follows the guidelines of a group.

Section 7.5 (part 1) due November 5

Can we go over Corollary 7.24 and review what right cosets are? And can you explain Lagrange's Theorem and the notation they use in the theorem?

The beginning of this is mostly review, so we have learned most of it before, applying it to rings and integers.

Section 7.4 due November 3

I don't understand the inner automorphism of G or Theorem 7.20.

It seems natural that this section is in here since there was a section on isomorphisms for rings.

Section 7.3 due Nov 1

I didn't understand the proof of Theorem 7.11. Many of the other theorems are obvious.

As in rings, it is helpful to know that G is a group so that you don't have to prove as many things, like associativity in subgroups.

Section 7.2 due 10/29

It says a^n=aaa...a (with n factors) but (ab)^n does not necessarily imply (a^n)(b^n). So does (ab)^n=(ab)(ab)(ab)(ab)...(ab) (with n factors of ab)? They said that we will use standard multiplicative notation, so shouldn't that mean that this is commutative?

The proof of part one of 7.5 was at first confusing but, upon further review, made a lot of sense and seemed extraordinarily creative. Then again, when are proofs not creative?

Section 7.1 (part 2) due 10/27

The section says that a nonzero ring is never a group because of the zero ring. Since multiplication is such a straightforward operation, why do they not make an exception in the group to say that for everything else, the group will work?

I thought the reflections were interesting because I haven't thought about reflections of shapes for a long time. I also think the theorems were somewhat interesting because they find some groups for us.

Section 7.1 (part 1) due Oct 25

With rings they say that if ab=ba then the ring is commutative. On p 163 they say that a group is abelian if it satisfies the commutativity axiom. Why is there a special word for it? Do they do it since multiplication commutativity is much more simple than what some operations can be? Can all groups' permuations be represented by an array?

Function composition is associative, but how often are composed functions commutative? Do they have to be simple, linear functions? That would be something interesting to explore a little bit. (Of course, they must not be too rare because there is a special name just for this type of group.)

Midterm course evaluation

1. The homework and the blogs contribute most to my learning. I like the blogs because it forces me to read the section before we talk about it in class. I think it helps you teach better because then you know what to spend more time talking about. The homework helps because it really is challenging and gives us a good idea of what kinds of questions might be on the test.

2. In class, it seems like we go over the proofs really fast. I feel like I am so busy taking notes that I am not taking everything in. (This doesn't seem to be true for the majority of the class, though, since they ask lots of questions.)

Review for test due Oct 20

I think ideals and quotient rings (and basically all of chapter 6) will be the most important topics that we have studied that will be on this test.

Ironically, I feel most uneasy about quotient rings. It seems like there is a lot to know about them (even though it is really only covered in one section) or at least a lot that can be proven from them, thus, it is necessary to study this more intensely for the test.

The question I would like worked out in class comes from 6.3 #11: Show that the principal ideal (x-1) in Z[x] is prime but not maximal.

Section 6.3 due 10/18

I don't understand the concept of a maximal ideal. And why can there be multiple maximals?

It is neat that all the concepts we have learned are interrelated and all the theorems relate these concepts. (Unfortunately, I don't understand some of these individual concepts so I can't yet see the bigger picture for 6.3.)

Section 6.2 (part 2) due 10/15

I don't fully understand the terms "natural homomorphism" or "homomorphic image." It says that if S is a homomorphic image of R then R and S might have the same algebraic properties.

It seems that this section is important because it relates things that can be helpful when writing proofs. It is interesting that the section says, "A concept developed with one idea in mind may have unexpected linkages with other important mathematical concepts. That is precisely the situation here." It's neat that there are other results than just what they had wanted.

Section 6.2 due 10/13

More than any other section in the book, I am having a hard time visualizing this (as much as Abstract Algebra can be visualized). In Theorem 6.8, are a, b, c, and d representing numbers? All of this new notation, including and especially "R/I", will take some getting used to.

If I understand this section accurately, then this is the method behind doing multiplication in my head. For example, if I={x in R 3y=x} and a=2, then 83=a+nx=2+(27)(3). Is this what they are saying?

Section 5.2 due 10/6

I don't understand it Theorem 5.7 when it talks about F* being an isomorphic subring of F[x]/(p(x)). How is F* different from F?

This is pretty straight forward because a lot of theorems about rings are now applied to polynomials, like the rules of modular arithmetic. However, there are some new theorems that, to me, don't seem as obvious as others, but will probably be helpful in proofs.

Sections 4.5 and 4.6 due 10/1

Theorem 4.29 says that if a polynomial in the reals is irreducible then it is either of degree one or two. In the proof of it, it says that this is not necessarily the case--that higher degrees have roots in the complex numbers. Because of the way it is stated in the theorem, though, it makes me wonder if I am understanding it correctly. Are there really only irreducible polynomials of degrees 1 and 2?

There are many theorems in 4.5 about primes, coefficients, etc. Some of them seem obvious while others seem brilliant. All of them make me wonder if they are applicable. (I suppose they are because they are in the textbook.)

Section 4.4 due 9/29

Is the difference between f(x) and F[x] that f(x) is the function of F[x], meaning that F[x] does nothing? When the corollaries are talking about a function having no roots, do they mean that they don't have real roots (but there exist complex roots)?

The most interesting part of the section was Theorem 4.14. At first I thought it meant you didn't have to do polynomial division--you just had to subtract f(a) from the polynomial--but then I realized that it just meant that it gives the remainder. Despite the disappointment, I always love how math works out perfectly.

Sections 4.2-4.3 due 9/27

If we had common divisors of f(x) and g(x) that were monic and had the same degree, how do we tell which one is greater? Is it the one with the greater coefficients? (For example, what is the gcd if the two divisors are (x+1) and (x-1)? Or would it be the product of the two if those were the divisors?)

Most of these theorems are the same as the theorems for the integers. The ones that are specifically geared toward polynomials are about ir/reducibles and the degrees of reducibles.

Section 4.1 due 9/24

I understood Theorem 4.1 until part iv--is it saying that the bi = 0R because the coefficients are equal to 0? I don't really understand part iv.

I like the familiarity of this section. I started learning about polynomials years ago (middle school or high school?). And the rules are still the same. I like dividing polynomials. Although this class is all about abstraction and proving things, which is not what I did with polynomials in high school, I think it is the start of what will be an interesting chapter.

Exam prep due 9/22

Some of the topics that we have studied that are of utmost importance are greatest common divisors, the Fundamental Theorem of Arithmetic, congruence modulo n, and the definitions of rings, identity, fields, etc.

I expect to see a lot of definitions that require specific details for full credit (after all, those details are crucial), proofs about the above topics, and proofs that we have done in class. I expect that it will be a fair grade of how much I have prepared for it, which will (hopefully) be a good thing!

Section 3.2 due 9/17

I don't understand the discussion on the zero divisor (or why it is important, rather) because it seems like only a unit would be important. Also, are there zero divisors in rings other than matrices and modular arithmetic?

I really like this section. You can never assume that all functions in a ring are valid--it has to be defined. BUT you can use previously shown axioms to show that things like subtraction work, which is awesome! Everything works out perfectly, which is what we, as mathematicians, want to happen.


Warning: Math joke below.

How does one insult a mathematician?

You say: "Your brain is smaller than any > 0!"

Section 3.1 (con't) due Sept 15

• I have spent about 3-4 hours on each homework assignment. The book is helpful, the lectures are more helpful (because they clarify what the theorems are saying), but they do not feel like preparation for the homework. Ultimately, I think I just need to keep at it to improve my creativity for the proofs. A lot of times, I get stuck on the proofs because I don't know what is sufficient or what the next step is.
• I think the things that would help me learn more effectively are things that are up to me: stop by your office hours and study with a group.
  

Section 3.1 due Sept 13

I don't understand the definition of an integral domain (where it says 1 sub r is not equal to 0 sub r and the discussion of the zero ring, in particular).

I like the idea of a ring--it is a subset that has solid properties. I like that the set of integers mod n is a ring. It seems like the concept of rings will be really useful in application and in proving things.

Section 2.3 due Sept 10

I don't really understand Theorem 2.8 and what the set of all congruence classes modulo n, with n being a prime number, has to do with ax=1.

The cool thing about this section is that some of the formulas look similar to those that I saw in Linear Algebra, such as Theorem 2.11. The difference is that those dealt with matrices but these equations deal with congruence classes.

Section 2.2 due Sept 8

The most confusing part of 2.2 was p 33 with the addition and multiplication tables.

To me, the properties of the classes were interesting because they are the same properties that we learned in Linear Algebra that must hold true for subspaces. As those are subspaces within a space, these are subsets within the integers.

Section 2.1 due on Sept 3

The most difficult part of the material was the concept of congruence classes. Theorem2.3 describes equal congruence classes, but aren't there only supposed to be n classes?

From what I do understand of congruence classes, it seems like a cool idea. It seems like it relates to the form that the division algorithm takes in because an integer is multiplied by something and then a remainder is added to get the desired integer.

1.1-1.3, due September 1

The most difficult part of the material was in 1.2 where divisibility was discussed. When they related the greatest common divisor with the smallest positive integer that can be written in the form of au+bv, I was confused. I had to re-read it a few times and try a few examples to understand what it was saying.

The most interesting part of the material was the Euclidean Algorithm. I learned it in Math 190, but I still find it intriguing that you can find the linear combination of two numbers that equals their gcd. I also find the Fundamental Theorem of Arithmetic interesting.

Introduction, due September 1

What is your year in school and major?

I am a senior majoring in math. (You may wonder why I haven't taken very many upper-level math classes--it's because I took a long time to decide what I wanted to major in. Now all I have left are math classes.)

Which post-calculus math courses have you taken?
I have taken Math 190 and 343.

Why are you taking this class?
I am taking this class because it is required for the math major.

Tell me about the math professor or teacher you have had who was the most and/or least effective. What did s/he do that worked so well/poorly?
The most effective math professor I have had seemed to really care for each person. Most college professors don't care to learn students' names, but she had ours memorized within the first two classes.

In class, she would teach us new material and then we had to teach it to the person sitting next to us in 2 minutes, which was effective because we had to pay attention so we could understand it. Different questions came up while we were teaching each other, so we would ask the professor and she would help us understand. This was very effective.

She also had plenty of time for office hours and for students to ask questions in class. I think it is important for the students to be comfortable asking questions, both in class and in office hours.

Write something interesting or unique about yourself.
Like I stated before, it took me a long time to decide to study math. I'm not a genius with math, but I find it very rewarding when I put a lot of time into it and understand it. I look forward to doing well in this class.

If you are unable to come to my scheduled office hours, what times would work for you?
I should be able to make it to your office hours on Tuesday and Thursday. If I needed help on other days, anytime between 10-1 would work on MWF.