8. 4 due April 8

This reading was neat. I was bugged at first that the book didn't offer proofs for the Sylow theorems but I didn't realize that they had a whole section for it. The hard part was actually reading through the proofs, but the weren't super difficult.

I'm interested in the idea of conjugacy equivalence. I wonder if groups can be constructed from conjugacy classes and if they would be any interesting.

7.10 Due March 30

This was definitely difficult reading. I got the two lemmas pretty well, but the proof for 7.52, which was the whole point of the section, was a beast. I look forward to going through the proof in class.

Since every finite group is isomorphic to a group of permutations, it seems like every group is going to have a normal subgroup isomorphic to An so the same theorems that we learned in this section would apply to all those subgroups.

7.9 Due March 27

The thing I mostly found difficult was seeing how any permutation can be written as the product of disjoint cycles, but then it made sense when I thought of how you could just see it as different move made on the identity permutation.

I think this is neat information and I wonder where it is applied the most. I wonder what kind of subgroups you could make somehow using the disjoint factorization of permutations.

Due March 25

Before the exam, I think I just need to work on understanding what I can do in manipulating a group or subgroup and what can't necessarily be done.

It would be great if you did another problem where you show what orders all the elements of a finite group must have.

7.7 Due March 20

I guess it figures that after talking about congruence in groups, it's only natural that we then find some analogy to quotient rings in group theory. I didn't find the concept hard to understand but the difficult part for me was working through the examples.

In pondering the material, the idea that comes to mind is that we aren't really going to be interested in finite quotient groups since they're just going to be isomorphic to some other finite, simpler group. Infinite quotient groups on the hand are probably a lot more interesting.

7.6 Due March 18

It was fairly difficult to understand at first why exactly we should care about this seemingly small part of groups, but after reading and thinking, I guess since we're trying to understand the structure of groups as much as possible, it's important.

I'm not sure how this applies to other things. I'm not sure whether we're going to be more interested in normal groups or in non normal groups, but I want to find out.

7.5 part 2 due March 16

The reading wasn't too difficult. It was already easy to tell that there would only be a finite number of groups and so classifying a few of them isn't that hard. I found it a little hard to verify that there was only two possible nonisomorphic groups of order 6, though.

I would be interested to look at the classification of finite fields. It seems like there would have to be a finite number of nonisomorphic fields of arbitrary order.

7.5 part 1 due March 13

The thing I had the hardest thing to understand for me was the concept of congruence in groups. I found it difficult because when we studied congruence in rings, both addition and multiplication were involved in the definition of congruence.

In thinking about the subject material, I think this chapter is important in the context of structure of groups which is the soul of group theory.

7.4 Due Wednesday March 11

Most of the chapter wasn't difficult. Since we've already extensively studied isomorphism of rings, it was easy to understand how two groups could be isomorphic. The last two theories, however were harder to wrap my head around.

In reflection, I wonder how interesting cyclic infinite groups. Since they're just isomorphic to Z, studying any infinite cyclic group should just be like studying Z.

7.2 Due March 6

The thing I found most difficult today was reading about the definition of a^(-1), because I've been using that notation on the homework for 7.1. So I was distressed that I'd been using it before showing uniqueness.

In thinking about the material, I think it's neat how by just knowing a few things basically constructs our group for us (commutativity, order of elements, etc)

7.1 Due March 2

Honestly, I don't think I found anything in the chapter to be hard, since I've been studying group theory independently for a while. The only thing that was new is that the definition of a group that I first learned did not include the unit axiom. I don't know what implications this has, but I'm interested to find out.

In thinking about finite groups and permutations, I figure there can only be a finite number of nonisomorphic groups with an arbitrary number of elements. I'd be interested to learn how to figure out the number of possible groups given a number of elements.

9.4 Due February 27

The hardest thing I found in the reading today was the last theorem of the section, theorem 9.31. I had to read it a couple of times before I understood what it was saying. And I still don't exactly why theorem matters. The book didn't seem too excited about it and din't offer any examples or anything.

The thing I reflected on most was how to use this same rigorous process to define fields of quotients on other integral domains besides Z. Like polynomial rings, congruence class rings and some of the more abstract quotient rings.

Due February 25

I think the most important things we've studied since the last test have been ideals and quotient rings and how quotient rings are the generalization of the other rings we've studied.

I also think that this last chapter we've done has been the most difficult and thus, is the one I need to study the most.

I would like to see a problem similar to #40 of 6.1 done in class, because I had a lot of trouble with that one.

6.3 Due February 23

This whole chapter has been pretty out there and I think this section has been the hardest to wrap my head around, but I really liked it just the same. The hardest part to understand would be the definition and why that makes a R/M a field. But looking at and thinking about the examples helped me more than anything.

In thinking about the section, I just wonder what kinds of ideals would be prime in some of the rings we've been looking at. Like the ring of continuous functions.

6.2 Due February 20

Well, I didn't have too much trouble in understanding the material. I guess it took me a little bit to realize how the natural homorphism pi is really just forming what is analogous to the rings of congruence classes we've been looking at with Zn and F[x]/p(x).

So I guess my extrapolation question would be what quotient rings would also be field, since some in Zn and F[x]/p(x) were and some weren't.

5.2 Due February 11

The only difficulty I had with this reading is seeing the importance of some of the latter theorems concerning whether F is a subring of F[x]/(p(x)), because my initial understanding of the concepts renders the theorem obvious since the constant polynomials in F[x] will still be distinct congruence classes and if you were only to look at their multiplication addition tables, you would have the same thing as F. It'd be nice to know if I'm missing something here...

In reflecting on the material, it appears that we can construct F[x]/(p(x)) to be a field itself provided p(x) is reducible, because then the gcd of p(x) and all the members of congruence classes will be 1.

5.1 Due February 9

The reading was fairly easy, because almost all the theorems were just previous congruence class arithmetic theorems applied to polynomial congruence class arithmetic. I guess the hardest part was really understanding corollary 5.5 and its implications.

In reflecting on the subject material, I just think about how complicated some of this arithmetic could potentially become. Especially if you consider rings of the form Z_n[x]/(p(x)) and have to kinds of modular arithmetic going on. Or what if you used that kind of ring to form another polynomial ring to make something like (Z_n[x]/(p(x)))[x]...

4.4 Due February 4

The only difficulty i had was being convinced of the remainder theorem. However, after reading the proof and trying out some out examples, I was convinced.

The only reflection on the reading that I had is that polynomial functions is really the same thing that we've been doing since middle school, only with a more formal approach. I'm curious as to what benefits one can gain from looking at the concept through these formal lens.

4.2 Due February 2

The thing that was most difficult for me was the part about if f(x)|g(x) then cf(x)|g(x). I find this weird because that's definitely not like normal arithmetic. The other thing that bugged me was why the gcd of two polynomials is the highest degree monic polynomial that divides both polynomials. This bugged me a little because of all the potential cf(x)'s that could be greater common divisors. However it makes sense, because we could have an infinite number of divisors and really we just need one to be defined as gcd which can be defined in all cases and the monic polynomial fits that role.

The thing that this reading made me think about is that if we can have relatively prime polynomials, can we have prime polynomials? How about unique prime factorization of polynomials? This is interesting stuff.

Due January 28

I think the most important things we've learned so far are the theorems involving modular arithmetic, especially those concerning solving modular algebra equations, and the concepts of isomorphism and homomorphism along with the axioms of showing whether two rings are isomorphic or homomorphic.

I imagine that the test will mostly include equations to solve in modular arithmetic and potential rings to test and figure out if they have identity or are commutative or integral domains or fields or isomorphic to another ring.

3.3 Due January 23

The hardest part of the reading for me was visualing some of the examples. For example, thinking of homomorphic rings that aren't isomorphic was a little difficult at first until I realized that the image of the function is a subring of the codomain and that the function is either injective or surjective. Also, I still can't think of any properties that are not preserved by isomorphism.

I think that the concept of isomorphism will make working with rings easier, especially rings constructed from finite sets. Because if the set is finite, there must also be a finite number of distinct, non-isomorphic rings from sets of the same order.

3.2 Due January 21.

So the hardest part for me in this reading was working out the proof that while every field is an integral domain, only finite integral domains are necessarily fields. But I do find it interesting that in order to determine whether a ring, finite or infinite, is both an integral domain and a field, all one has to do is show that it is a field.

As far as reflection on the reading goes, I just think it's really cool that subtraction and exponential operations work on rings. Also, in thinking about division on rings, I think that division would be defined on a ring with multiplicative inverses.

3.1 Due January 16

Oops, I forgot to blog for last time. Oh, well. Now, this last reading was great. The most difficult thing for me was probably just checking all the examples to make sure they fulfilled all the axioms. That took a while.

What I found interesting is that the set of all 2 x 2 matrices of the form listed in one of the examples forms a field while the set of 2 x 2 matrices. This makes me wonder what other forms of 2 x 2 matrices farm fields and also how one would form a field with 4 x 4 and 3 x 3 matrices using the standard definitions of matrix addition and multiplication.

2.2 Due January 12

Well, this reading was slightly more difficult on a conceptual level than the previous readings. It was a little difficult to wrap my mind around the idea of adding and multiplying whole congruence classes instead of integers.

Over the Christmas break and in preparation for this class, I started reading up on group theory. One thing that I noticed in reading section 2.2 is that addition and multiplication of congruence classes can be used to define a group on Z_n.

2.1 Due January 9

Once again, the reading was strictly review of material from Math 190 and I've done a lot with modular arithmetic outside of 190 anyway, so I feel very comfortable with concepts in today's reading. I didn't feel challenged at all, but I did come up with a question. The book showed that that addition and multiplication still work as normal in modular arithmetic, but I wonder if the same goes for exponential manipulation. I plan on exploring this question.

I was impressed on how the use of modular arithmetic could make some of the exercises of the last section much easier. For example, one of the exercises in 1.2 was to prove that a positive integer is divisible by 3 if and only if its digits add up to a multiple of three. This proof would have been much easier with modular arithmetic because it's clear that powers of 10 are congruent to 1 mod 3 and thus it's easy to show that the sum of the digits of any number is congruent to the number itself mod 3.

1.1-1.3, due on January 7

To be honest, I wasn't really challenged by the reading assignment. I have just recently completed Math 190 so all of the reading was review. However, it did take me a little time to completely understand the proof of the division algorithm.

I love the idea of unique prime factorization. I'm really impressed on how it has increased my understanding of algebra concepts, especially those concerning divisibility. And understanding it really helped me on the exercises of the earlier sections, interestingly enough.

Introduction, due on January 7

What is your year in school and major? I'm a sophomore and a mathematics major.
Which post-calculus math courses have you taken? Math 190, Math 214, Math 334 and Math343.
Why are you taking this class? (Be specific.) For my major, but more importantly because it sounds really cool.
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? I think Dr. Doud here at BYU has been the most effective teacher that I've had. He did everything in a very orderly manner and always stayed true to his word. Also, he presented things very logically and clearly.
Write something interesting or unique about yourself. Well, I've worked three summers as a river guide on the slmon river in Idaho. Also, I served a mission in Albania and now teach Albanian at the MTC.
If you are unable to come to my scheduled office hours, what times would work for you? Any time on Tuesdays and Thursdays.