final post, due April 13

1st question
Trying to choose which topics/theorems are important? I would assume all of them, but mostly rings, groups, ideals, quotient rings, quotient groups. Also, Lagrange's Theorem, Cauchy's Theorem, Division algorithm, polynomials, congruence classes.

2nd question
I need to understand the finite abelian groups and sylow theorems as well as direct products...basically chapter 8 is scary to me. A mathematical question: how do you compose k-cycles? I'm sure I did those wrong.

3rd question
The things I learned will most likely not be used directly after this semester. However, like any math course, this class has helped me become better at proving and thinking. Also, it has been interesting. So I will take from this course a greater appreciation for algebra and a sharpened intellect and ability to prove things--which I value very highly. It's been a good course, though tough at times.

Section 8.3, due April 12

What was most difficult?
The most difficult thing is keeping the definitions straight, because there are definitions embedded within definitions. Let me give an example: A sylow p-subgroup is a subgroup of order p^n when p^n is the largest power of p that divides |G|. So, to know what a slow p-subgoup is, you have to wrap your head (in my case, not a smart head) around this definition. Now, In Corollary 8.16, we reference the name sylow p-subgroup, but then I have to embed that definition in 8.16 because I still havnen't wrapped my head around what a sylow p-subgroup is. This is a common thing in mathematics, but when all the terms are new to you, it gets confusing. I have to keep referring back to the previous page. It is a whimpy complaint, I know.

Reflections
Not much to reflect about on this, but I was at the MAA conference on Friday, and a fellow BYU student, Kyle Stewart, gave a presentation in representation theory. He used lots of the vocabulary we have covered in this class. Even though I couldn't follow everything he was saying, I was able to a little bit because of this course.
That was fun!

Section 8.2, due April 8

What is the most difficult part?
The theorems are straightforward, the proofs are the same (but a bit boring). The difficult things is finding relevance right now. I feel like we've beat groups to death, which I know isn't the case. I'm sure there's MUCH much more out there.

Reflection
I've been way busy, getting ready for the MAA conference (at which I'm presenting) tomorrow. My computer crashed, and being an idiot, I didn't have back up, so I've spent the last two days working on that. Tomorrow, I'll be there, so I won't be in class (again). I feel bad and disrespectful to you for missing class, but I definitely haven't been sitting at home. 17 credits and trying to graduate...yuck. Things got crazy at the end.

Section 8.1, due April 6

What was most difficult?
After reading this section, I asked myself, "what was that all about?", which is not a good sign!
let me try and summarize what I DO understand:
the direct product is the cartesian product of several groups, and is itself a group.
but a particular group is not a subgroup of the direct product.
Normal subgroups whose intersection is the cyclic group of the identity have abelian elements, where an element is from each group.

it's a bit clearer what I don't understand: the 2nd example on pg. 245, specifically when it moves to page 246, theorem 8.1, and theorem 8.3

Reflections
Just trying to stay alive with all my classes, and MAA conference (I'm presenting!) on Friday.

Section 7.10, due april 4

What was the most difficult part?
The most difficult part was the proof of theorem 7.52. It was long and I found it difficult to pay attention the details.

ReflectionI'm looking forward to getting to a new chapter and to see some "other topics in group theory"

Section 7.9 due April 1

What was most difficult?
Theorems 7.50 and 7.51 just seem a little vague, but I'm confident it will clear once I've done some exercises with them.

Reflection
I have been swamped with computer science lately, and it's made it way hard to attend class. When you have projects that are time intensive, or exams to study for (both of which have been constant the last week and a half, including 5 exams in one week(!), it comes down to, "study for this thing, or go to lecture?" and I feel bad about missing lecture. It also makes the homework much harder. I should be able to go tomorrow though, as things have let up in my other classes...CS is still busy though. *sigh*

Section 7.8, due march 30

What was most difficult?
It's hard to know...I read this too fast. I am so swamped with 17 credits and tests and assignments, and my job, that I keep putting these readings off until the last minute (or last five minutes). I can usually find something interesting, but not tonight...I need to be a bit more organized this last stretch, but it's hard with so much going on.

Reflection
Where is the Second Isomorphism Theorem for Groups?

Section 7.7, due March 28

What was most difficult?
The most difficult part of this is just trying to grasp it without having worked with it or seen it in lecture yet. I mean, what I'm really saying is, it's hard to know what will be difficult until I start playing around with it. You can tell by the time of writing that I don't have time to play around with it right now. I thought quotient rings were a bit annoying or difficult to understand--the hw problems were certainly harder--and I hope quotient groups aren't quite as bad.

Reflection
This doesn't seem too bad. We'll see how things go these next couple of days.

The rest of 7.6, due March 25

What was most difficult?
The bold print at the top of page 212 seemed a bit confusing at first, I didn't know what it meant by saying Na=aN does not imply an=na For all n in N. But the example right below cleared it up.

ReflectionJust glad the exam is over and that we are moving forward to the end of the semester!

Section 7.6, first part, due March 23

What was most difficult?
The most difficult part of this section was focusing! I'm finishing my preparations for the exam, so it's hard to be thinking of new material. I thought it would be difficult to deal with right and left cosets for these subgroups, but once I read the examples after the definition of normal subgroups, it seems pretty straightforward and manageable.

Reflection
The best thing about this section, was that it cleared up cosets for me. For some reason, I was struggling in the last section with the assignment. Using the definition as exactly as I thought it was written, I kept getting the wrong idea for cosets. This section made it totally clear.

Pre-exam writing, due March 21

What topics/theorems are most important?
I think quotient rings, ideals, and groups, especially congruence in those areas, are the most important concepts learned in class since the last exam.

What kind of questions do I expect to see?
Based on the last exam, I expect to see some conjectures that are either true or false, and require our justification. I also expect to see some proofs of some of the named theorems from class.

What do I need to work on understanding?
I really need to review things like cosets and cyclic subgroups.
A question: List all the cyclic subgroups of D_{4}
I had a hard time seeing how this definition fit in with this idea, because it seemed as though all the rotations in D_{4} can be done several times...there is something I'm not understanding about this definition.

Section 7.5 part 2, due March 18

What was most difficult?
Nothing was difficult in this section, at least I don't think so. I feel like it's really hard to judge this until I've done the problems, which doesn't happen yet because it's not due for two more class periods, as I'm working on 7.5 1st part right now. I'm having some difficulties in the previous section, namely with right cosets.

Reflection
I'm starting to feel more dumb, and less good at algebra as this class continues. It's probably a combination of being very busy, this material being totally new, feeling tired, and a little bit of senioritis.

Section 7.4, due March 14

What was most difficult?
The most difficult thing for me is cyclic
Groups, covered in 7.3, but referenced here in the context of
Isomorphism

Reflections
Not much to reflect on.I guess i usually
Find it fun to show somethng is an isomorphism, so i'm looking forward to the
homework assignmwent.

Section 7.3, due March 11

What was most difficult?
Theorem 7.15 has a confusing hypothesis....multiplicative group of nonzero elements of a field? Wolfram helped a bit, but it was just kind of wordy.

Reflection
I always enjoy it when we do the subspace, subring, subroup, etc. There are less axioms to prove!

Section 7.2, due March 9

What was most difficult?
The most difficult thing is just keeping things straight between rings and groups. Nothing too major, just gotta get the definitions down pat.

Reflection
I like dealing with groups much better than dealing with quotient rings!

Section 7.1, due March 4

What was most difficult?
Nothing was too difficult about this. It is pretty straightforward they way they are laying out the function as an array and composing it. Hopefully this gets easier to understand than the last bit about quotient rings and what not.

Reflection
Excited for a new section!

Section 6.3, due March 2

What was most difficult?
The most difficult thing right now is...everything. The text is getting less and less clear, or I'm getting more and more stupid. I think really though, it's just understanding why we are abstracting these things that is a bit unclear and makes it hard to grasp this. It was pretty clear in the earlier sections why were doing this abstraction...but things like maximal ideals? What is it for? I don't care if it's used in the "real world" how is it used in algebra?

Most Interesting
I think it's interesting that every maximal ideal in a commutative ring with identity is prime.

Section 6.1 (1st part), due February 25

What was most difficult?
The most difficult thing is some of the notational characteristics, like the top of page 146 and in general, this stuff is just getting a bit hard to keep track of...abstractions, definitions, etc.

Reflections
What the heck is meant by the first example on page 146?! I hope I don't get murdered on the test...

Section 6.1 (2nd time), due Feb 23

What was most difficult?
The most difficult thing was (nothing new), just the new vocab and I'm not sure I like the notation a+I...I think in general the notation in this subject can be a bit misleading.

Reflection
Nothing is super difficult, it's just a pain to have class two days in a row! But at least we had a long weekend to do the previous assignment (of course I didn't use my time effectively...long weekends encourage relaxation...bad news).

Section 6.1, due February 22

What was most difficult?
The most difficult thing is understanding the bigger picture of what we're doing, and why ideals are relevant. The next most difficult thing is having to remember and understand more, very specific definitions.

Reflection
We'll see how things go in class... I'm interested to learn about ideals, but also scared because supposedly things are going to get harder yet before they get easier, though class hasn't been too hard just yet.

Section 5.3 Due, February 18

What was most difficult?
The most difficult part is still understanding modular arithmetic in these fields.
The last assignment wasn't very clear or helpful. Also, I missed lecture Wed so I didn't get any good info. I'm hoping tomorrow's lecture can help.

Reflection
I thought the discussion about complex numbers was kind of cool. I think Complex Analysis (well, parts of it) is pretty cool so it was just neat to see we could get the set of complex numbers buy using these fields.

Section 5.2, due February 16

What was most difficult?
The most difficult part was the paragraph at the top of page 126...I just feel lost right there.

Reflection
I have been super, super busy in other classes, with tests and projects so it's been hard to keep up the reading for the last two days. Things are just pretty stressful and I hate having to choose between sleep/my health and keeping my head above water. After tomorrow things should settle down and hopefully I can get into the groove I was in before the test for this class.

Section 9.4, due February 11

What was most difficult?
The thing that was most difficult was the notation [a,b] for a/b, because
I'm so used to the latter notation. However, towards the end of the section they switched, which made things easier. In general, what has been difficult and further exemplified in this reading, was how they took something so intuitive and give us these super specific, strange names for everything. This makes me over-think everything and when I try to remember the definition, I feel confused until I remember the aspects that are more in line with my intuition. But, this is mathematics, so it's no real surprise, just a bit obnoxious at times.

Reflection
It will be fun/interesting to do some of what we've been doing, but with rational numbers now.

February 9 Reading Assignment

Topics/Theorems I think are most important
I think rings and polynomials are most important, and the named theorems associated with each. Also, the counting in Z mod n is important, but that's pretty easy so I'm not too worried about it. The Division Algorithm (also the DA in F[x]) will be important, perhaps most importantly for using it prove things. The Remainder Theorem and Factor Theorem will be important.

What kind of questions do I expect to see?
I expect (and hope) the questions are similar to those on the homework, which are usually very straightforward, though they require a thorough understanding of how the theorems and corollaries can be used to find new results. I expect questions such as prove...(blah blah blah) is irreducible, or find the linear combination of the gcd of these two numbers (or polynomials), etc.

What do I need to work on better before the exam?
I need to memorize the definitions better, and be able to command them. Some of them right now, I still have to refer to the text, so I need to get those down pat and understand the nuances and what is what so I can move through the material a bit more confidently. I feel like I know most of the other stuff pretty well, it's just that I forget the EXACT definition at times.

Section 4.4, due February 7

What was most difficult?
I have a hard time choosing what was most difficult because this section was pretty straightforward, especially when compared to the last section. Friday's lecture cleared that up though and I'd be lying if I was to make up something from this reading that was hard.

Reflection
I think there are some cool theorems in this section, namely The Remainder Theorem and The Factor Theorem. I had just never really thought about finding the remainder by evaluating the function at the point a from the factor (x-a).
Surprisingly, I am thinking this class is pretty cool. Even though I'm more of an applied mathematician, and analysis is a bit more directly related, I HATE analysis and I think Algebra is much more interesting and fun, so far at least. At least Algebra is doable by HUMANS, and the hw isn't impossible to figure out, though it is challenging enough.

What was was most difficult?
The thing that is most difficult right now is that there are so many new terms and concepts...they are just piling on and when I'm doing the reading, I feel like my head is going to explode and I don't have time to process it all. Reading just makes me look forward to lectures so much more.

Reflection
As mentioned, the reading is confusing, but once I've been to lecture and when working on problems, it's not really that bad.

Recap-due January 28

How long have you spent on the HW assignments? Did the lecture/reading prepare you for it?
I usually spend about 1.5-2.5 hours on each assignment. The lecture is usually more helpful than the reading. I'll be honest, the blogging is a bit annoying, BUT, it is forcing me to read before the lecture, so it is a very smart strategy on your part. I guess overall I really appreciate it. I think the reading can be somewhat v--like I don't know what questions to have--and I find myself going back and reading it again when I'm working on the problems. I think this is just how mathematics is--it's hard to know what's going on until you are DOING the work.

What has contributed most to your learning in this class thus far?
The most important aspect of my learning has been doing the problems. It's thinking over them, struggling with them, sleeping on them and then asking for help from classmates that helps the most. I don't want to make the assignments sound harder than they are. But, usually there are one or two problems on an assignment that are a bit tricky and take some extra time.

What do you think would help you learn more effectively or make the class better for you?
Maybe reading a bit slower would make doing the problems a bit easier. It's difficult because I have so much to do that I always feel pressured to get the assignments/reading done as fast as possible, but this stifles learning. I feel that I learn best if I'm not rushed and am a bit more relaxed. Maybe I could achieve this by blocking out a little more time so I'm not so pressed. I think the lectures are going great. Not trying to brown-nose, but you (Dr. Jenkins) are definitely one of the better lecturers I've had in this department, which I'm really grateful for. I wouldn't change anything about the lectures.

3.3 due on January 26

What was most difficult?
The most confusing thing for me was on the first example, the addition and multiplication tables are a bit confusing. Namely, why is it arranged 0,6,2,8,4? Is there a special reason or is the author just showing that it doesn't matter which order we make the table? It threw me off.

Reflection
I find myself wondering, "what is the point of this?" What is this leading up to/why do we care about this? Is this "useful" or is this just a mathematician's playground to make up rules for the sake of doing it?

3.2, due on January 24

What was most difficult?
The most difficult thing about this section was some of the new vocabulary: unit, and zero divisor. For unit, it just seems like a strange name. We have an element a that is a unit if some other number is it's reciprocal? That's sort of the intuitive definition I'm getting. And for zero divisor, we have a nonzero a where this is also a nonzero b such that their product is 0. That's not too difficult. It's still all unclear where this is leading.

Reflection
All I can say is I can't wait for Monday's lecture, to see this explained. It's usually easier to understand when you (Dr. Jenkins) explain it.

3.1 (through page 48), due on January 19.

What was most difficult for you?
In all honesty, this wasn't difficult. I've never seen rings before, but it was very clearly defined in the text. I've had enough other math classes, that the idea of constructing sets to meet certain properties(such as those of rings or fields) isn't anything new to me. The examples were very clear, and it became obvious to me before I read the example on real matrices that they would be considered rings as well. I don't know if this is a satisfactory answer, but nothing was difficult. The "most difficult" part of the text would equally be any part.

Reflection
I'm eager to see what we will do with rings, and why they exist. What is their role in mathematics? As an applied math person, I'm always interested in the utility of different branches of mathematics, and I believe (or want to believe) that in their own way, each part is useful. Therefore, I'm very curious to learn more about rings and their place in Algebra and mathematics in general.

2.3, due on January 14

What was difficult?
I'm struggling with the difference between corollary 2.9 and 2.10. We were told in the previous section that when it is understood we are working in Zn that we omit the brackets for the congruence classes. So, other than the fact that one of these has brackets and one doesn't, they seem to be concluding the same thing.AHHH...As I just wrote that I glanced again at the page and read again more carefully to see that for 2.10, n need not be prime...Got it.

Reflection
I was glad this section was short and that there wasn't much to read. I don't really have much to say about this topic in particular yet... I feel like I'd have more to say after doing the problems. I'm still just taking all of this (Abstract Algebra) in and seeing where it goes.

2.2, due on January 12

What was the most difficult part of the material for you?
The most difficult part is just wrapping my head around the idea of modular arithmetic. I guess the concept isn't really that difficult, but I'm not thrilled about the idea of having to remember more definitions for something that isn't very intuitive to me. Perhaps the new notation will be confusing. All in all, nothing is super difficult in the class yet. I'm sure it will get worse (or better, depending on your perspective) later. As for now, it's mostly just that the stuff is new.

Reflection
Theorem 2.7, or the different rules for the integers applied to Zn is pretty cool. It's just cool how in math, things can be generalized to work in different areas so many times--yet other times they can't.

2.1, due on January 10

What was the most difficult part of the material for me?
The Most difficult part of the material is the main idea of the section, congruence classes. I'm sure it will be seen later, but I have no idea what these are used for, which makes it difficult to understand it right now. So to me right now, it is just an(vague)idea. I'm sure once we have the lecture it will be a bit more clear. It's hard to say what exactly is difficult about this, other than I'm wondering why we care about this.

Reflection
I'd say the most interesting thing about this is that it's just so different than what I'm used to dealing with (this whole class is). I remember discussing congruence classes and mod for a couple weeks in Math 190, but even then we just learned how to solve the problems given to us--we weren't exposed to anything done at a higher level with it. That reminds me, what is mod and why do we care about it? Basically, I don't have time to read ahead in the textbook because I need to stay caught up in other classes, but I'm actually a bit excited to see what will come. I'm used to working with partial differential equations, matrices, etc. Something like this is a whole other branch of math of which I know nothing, and this section reminded me of that.

1.1-1.3, due on January 7

What was the most difficult part of the material?
This material isn't too difficult, as it's basically a review of Math 190 and some common sense stuff. I'd say though that the proof of Theorem 1.1 was kind of long, and as simple as it is, I probably wouldn't know how to come up with it on my own. But in all honesty, the topics covered in these first sections had easy to follow proofs and were topics that I'm already familiar with. I'm trying to think of this on a deeper level and I'm not sure I can.

Write something reflective
The most interesting part of the reading was the proof of Theorem 1.10. It was just a cute, clever proof by contradiction. I also enjoyed Theorem 1.3 because the talking of linear combinations reminded me of linear algebra, a subject I really enjoy.

Introduction, due on January 7

Answers to questions from the syllabus:

Q: What is your year in school and major?
A: I am a senior and my major is mathematics.

Q: Which post-calculus math courses have you taken?
A: I have taken Math 334, 341, 570, 521, 447, and Math 352.

Q: Why are you taking this class?
A: I am taking this class because it is required for the mathematics major.

Q: 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?
A: Dr. Barrett is the most effective I've had thus far. I had him for Math 570, a class I really loved. He was effective because his lecture notes were very well organized, he wrote clearly on the board at a comfortable pace, and worked very hard to help students have their questions answered. He's also a very kind and funny man, which helped me feel at ease about approaching him and discussing the course with him. Another way he was really effective was in his tests. I feel like the questions he put on exams were very well correlated to what we learned--no surprises. Yet, the questions were challenging enough that they required complete mastery of the material. Also, given the nature of the course and all the cool theorems and results associated with matrices, some of my most joyful moments in math were when I'd figure out how to prove or disprove something I'd been working on for hours, and the result or proof of how to do it was beautiful and clever. I guess I'm saying that he chose test questions that were actually interesting...something that I feel most people would want to know the answer to and be delighted as to why it is true or not. This is in contrast to teachers who've asked questions that are enormously beyond the scope of what was discussed in homework or previous preparation for the exam. I won't name names, but a really good professor who I think writes terrible tests did just that--he'd ask students to reach (I felt) too far. That sounds like I'm just complaining about hard tests...I'm not, because I have aced some of them, but objectively, I felt like I could prepare as best I could from homework and study sessions, etc. and then only hope that I'd figure out the result he wanted that was beyond the scope of the learning outcomes. I resented this. Other teachers have been ineffective by not being polite, talking in a monotone voice, or showing a lack of interest in the course or their students.

Q: Write something interesting or unique about yourself.
A: I am a pretty good drummer. I'm not in a band, but I love rock music and really enjoy playing.

Q: If you are unable to come to my scheduled office hours, what times would work for you?
A: I actually can't make office hours because I have a class then. A time that would work would be between 10 and 2 on Tuesday, or 9 am MWF.