Which topics and thms do you think are important out of those we have studied?

I would say that the isomorphism theorems were used quite a lot and seemed to impact a lot of other topics and theorems. I also think I've used Lagrange's theorem a lot, so it must be important.

What do you need to work on understanding better before the exam?

I still struggle with cyclic groups and the order of elements. When I work on filling out the study guide, I also usually have the hardest time finding enough examples of all the different things.

How do you think the things you learned in this course might be useful to you in the future?

I'll be honest. I don't really think it will be. And perhaps that is because I barely understand most of what we talked about in class, which makes it hard to apply it to things outside of the course. Additionally, I'm almost done with my Math Education degree and realistically I won't be getting a job after I graduate because I have 2 kids. I'm just trying to make it through and pass so I can graduate! I found that it was hard to find applications to teacher education for some of the advanced math courses that are required for the MthEd degree.

8.3, Due April 11

Difficult:

I was confused about the statement that if K is a Sylow p-subgroup of G, then so is x^-1Kx. Perhaps this is related to confusion about 'the image' of something. Were they just trying to say that a subgroup and it's image are isomorphic?

Reflective:

I noticed that Cauchy's thm was in this section - so we must have done it early in class and not with section 8.3. Why is that?

8.2, Due April 8

Difficult:

Oh boy. I really struggle with the "order" idea. I think it's very confusing that they have orders of groups relating to the number of elements, but then there are orders of elements and sometimes groups ARE elements of another group and it just all blends together. I have a hard time separating them and remembering which one we are talking about.

Reflective:

I thought it was helpful to have the little dictionary for translating from multiplicative to additive notation. I think we have seen some of it before, but I thought it was good that they reprinted it when it would be more usable for us.

8.1, Due April 6

Difficult:

For Lemma 8.2, I think I am confused. What is ? The cyclic subgroup generated by e? How is this different from the whole group? I think I've had this question before...

Reflective:

I thought it was cool that we looked at the cartesian product of more than two groups. I think it would be a little hard to keep track of what the elements actually look like, but it was cool anyways :)

7.10, Due April 4

Difficult:

Maybe it was because it was at the end of 7.9, and because we didn't cover it in class, but I don't feel like I understand what an alternating group is. Because of that, it's really hard to understand the theorems from 7.10.

Reflective:

I just find it interesting that we have a single-sentence theorem, and it takes 2 lemmas and 2 pages of textbook just to prove it. I really really hope we aren't expected to state & prove that :O

7.9, Due April 1

Difficult:

I don't understand why we would factor permutations down. Why would we care to write the identity permutation as (12)(12)? Why is every transposition it's own inverse? Does that mean every element has order 2?

Reflective:

I think it's cool that we are cutting down on the notation for symmetric groups, I'm just worried that I will get confused when I try to follow the composition through.

7.8, Due March 30

Difficult:

I think I am still confused on cyclic groups and generators of groups. When they say does that mean the set of elements generated by e? Wouldn't that be the whole group? Or is it just e because when you add(mult) e, you just keep getting e over and over?

Reflective:

I thought it was interesting that instead of the kernel being the set of things that are mapped to zero, it's the set of things that are mapped to the identity. In an additive group like Z, it would be the same thing, but in say the multiplicative group of units of Z8 it would be 1.

7.7, Due March 28

Difficult:

I'm not sure I understand how thm 7.36 works in regards to the order of G/N. I think we established a normal subgroup of order 4 in D4 today. So D4/N would have order 2? How do I know what 2 elements are in there?

Reflective:

I thought it was cool how they related the quotient ring R/I to quotient groups and how R/I was really just the additive group of the quotient ring R/I.

7.6 Part 2, Due March 25

Difficult:

It was confusing when they said that you can conclude from Na=aN is that if na is in Na then na is also an element of aN, so that there is t in N such that an=at in aN. Does that mean that n=t?

Reflective:

I thought it was cool how they connected normal groups to abelian groups and centers of groups, and it was good that they made sure to mention that it doesn't imply that the normal subgroup N is commutative.

7.6 Part 1, Due March 23

Difficult:

I got very confused when they were talking about applying other definitions of congruence to groups. It was especially confusing when they were trying to explain/translate congruence in rings under addition with congruence in groups under multiplication.

Reflective:

I thought it was funny how they had to develop a whole new congruence for the left side, and go back and rename what we called "congruence" to be "right congruence".

Exam Questions, Due March 21

Which topics and theorems do you think are the most important out of those we have studied?

Obviously any theorems with names (1st Isomorphism Thm, Lagrange's Thm) are important. It seems that the basic ideas on this test will be ideals, kernals, groups, subgroups, and centers of groups.

What kinds of questions do you expect to see on the exam?

I kind of answered that above. I'd hope it would be like the last test, with some definitions, some questions asking for examples, a proof, and an application of some theorems to something we haven't worked specifically on.

What do you need to work on understanding better before the exam?

I am not sure yet... I think I get confused with ideals and cosets and subgroups, and how they are different. I also find the order of an element confusing, and what it means for a group to be generated by two elements.

7.5 part 2, Due March 18

Difficult:

I had a hard time following the proof of thm 7.29 at first, when they were talking about how they knew ab=c. The other thing that I'm still confused about is that the theorems use "either or" language. Does that mean it doesn't matter which one we choose?

Reflective:

I think thm 7.28 would have been a lot more confusing if we hadn't talked about Lagrange's Theorem in class. I liked how Dr. Doud put it - if we have a group of order 24 and 6 distinct cosets, how many elements are in each coset? Obviously for prime numbers, it's either 1 or p.

7.5 part 1, Due March 16

Difficult:

I think the right coset idea is confusing for me. I know it came up before, but we never really addressed it so I haven't thought about it at all, but now it's in some of the theorems. I'm hoping we cover this in class so that I don't have to go searching through the book for it.

Reflective:

I thought LaGrange's theorem was an interesting idea but I'm not sure how knowing that will actually help me with anything. How is it significant that the order of a subgroup will divide the order of a group?

7.4, Due Pi Day

Difficult:

I still have a hard time understanding what an isomorphism is... I also think it might be hard to remember how the homomorphism part changes based on the 2 different operations that G and H could have.

Reflective:

For rings, there were 2 parts to a homomorphism, but for groups there is only 1. I suppose that's because there is only 1 operation on a group, it could be addition or multiplication (or something else that we define).

7.3, Due March 11

Difficult:

I got a bit confused after thm 7.13 when they started talking about cyclic groups. One thing they said was that every cyclic group is abelian, but I don't see why that would be the case.

Reflective:

Subgroups of groups follow along with the same idea as subrings of rings. It's a subgroup/subring if it is itself a group/ring under the same operations of the bigger group/ring. I also thought it was interesting that we only needed to check 2 of the 4 axioms because the identity comes naturally from those 2, and the associativity will be there.

7.2, Due March 9

Difficult:

I think the hardest part for me to follow was Thm 7.8 when they talked about order and finite order. I kept needing to refer back to the definition of "finite" order and try to make sense of statement 2. I'm not sure I understand why it happens the way it does.

Reflective:

I felt that most of the theorems presented in this section were easy to follow, and were just basic ideas that I was used to seeing, such as that a^m*a^n = a^(m+n). It was nice seeing some of those easy things again :)

7.1 Part 2, Due March 7

Difficult:

The previous homework, and the reading of the rest of this section didn't seem too hard to understand. The only thing that was frustrating was that they said to assume that unless said otherwise, we are thinking of the group operation as addition. Yet in the homework problems, they almost all dealt with multiplication.

Reflective:

I thought it was cool that we basically came up with Thm 7.1 in class. It was easy to see the relationship between groups and rings and try to figure out whether all rings are also groups, and so on.

7.1 Part 1, Due March 4

Difficult:

It seems strange to me to list the permutation the way they were doing it in the first example, but that may just be due to the fact that to me, it looks like matrices and would be confusing. I suppose It's basically set up like an x,y chart.

Reflective:

I'm so glad to be done with ring and field stuff. I'm hoping it's true what everyone has been saying, that group theory is much easier than ring theory. So far this section hasn't had any theorems or proofs, just definitions. There are a lot less axioms for a group than there was for a ring.

6.3, Due March 2

Difficult:

I don't understand their example at the top of pg 155. They say that the principle ideal (x) in Z[x] is prime, but since the quotient ring Z[x]/(x) is isomorphic to Z, then Z[x]/(x) is an integral domain but not a field. How does that isomorphism make it so that it can't be a field?

Reflective:

I think the definition (and explanation) of a prime ideal was the only thing that made complete sense in this chapter. It made logical sense to me to go from saying that when p|bc then p|b or p|c to saying when bc is in (p) then either b is in (p) or c is in (p).

6.2 part 2, Due February 28

Difficult:

I had a hard time understanding homo/isomorphisms previously... I can't imagine this is going to be any different. Now we're adding in the idea of a kernel, which I also am confused about. The kernel is basically just a zero map from what I understand of the chapter, but the way they sometimes talk about it makes me wonder if that's really what it is.

Reflective:

I think I'm in luck today! It usually happens that I do my blog 2 days before it's due, before class. As in, today we're talking about the FIRST part of 6.2 and I'm already doing the blog for the last half. Sometimes it can mess me up a bit, but today I'm in luck. Question #25 from homework 6.1 is proving thm 6.10! Now I can check my work and fix any mistakes :)

6.2 part 1, Due February 25

Difficult:

I think what will be the hardest to work with is the notation of addition of cosets. Just looking at one of the simpler examples of an addition and multiplication table on page 146 got me worried about what I would need to do with larger fields...

Reflective:

The addition and multiplication seems to be similar to that in Zn, but now we have to add a "+I" at the end. As long as I'm correct about that, it shouldn't be too hard to do that for all the new multiplication and addition that I need to do...

6.1 Part 2, Due February 23

Difficult:

At the top of page 140 when the describe the congruence class of a modulo I, I got confused. Their first set up with b-a in I makes sense. I don't understand why they carried on further from there... Why was it important to get to a+i such that i is in I? Why did we even need to develop that i in the first place?

Reflective:

This is basically like the congruence in Z, except that they added the idea of cosets and that instead of using [a] to denote the congruence class of a, they say a+I instead.

6.1 part 1, Due February 22

Difficult:

The concepts really aren't too difficult to understand, I'm just concerned that the application will be more difficult than it seems. Generally (and this section is no exception), I have to break down each theorem into it's individual parts and think "ok, so if a subring I of R is an ideal, then there's an element of R and an element of I, and when I multiply them together the product is in I. Ok, that makes sense."

Reflective:

I thought it was really cool when they talked about how the congruence modulo 3 can be characterized as a-b being a multiple of 3. It's really the same thing we've said all along, but either they haven't phrased it that way, or I missed it when they did. That definition makes a lot more sense in my mind than saying that 3 divides a-b.

5.3, Due February 18

Difficult:

I had a little bit of a hard time understanding the idea of the extension field K. The proof of 5.12 wasn't too hard to understand, but I had to break it down a little bit farther and think about how if p(x) is an irreducible factor of f(x) and p(x) has roots, then any roots it has are also roots of f(x).

Reflective:

Well I feel like this is what I've been saying this whole chapter, but this is a lot like the stuff we did back in chapter 2. What's cool though is that while F[x]/p(x) is a field just like how Zp is a field, F[x]/(p(x)) also contains a root of p(x), which is obviously not possible in Zp.

5.2, Due February 16

Difficult:

In the proof of Theorem 5.7, they make a set F* of the constant polynomials and verify that it is a subring of F[x]/(p(x)). What I don't understand is why don't they just state it that way in the theorem? Why don't they just say that the set of constant polynomials make a set F* which is a subring of F[x]/(p(x))?

Reflective:

Again, many of the proofs are just altered versions of the proofs for rings Zn, so the concept presented are easily recognizable.

5.1, Due February 14

Difficult:

Oh boy. And I thought regular modular arithmetic in Zn was rough. I still find the first chapter of that confusing, when they talk about things like 17=5(mod6). It's really not any different here in F[x] modulos except that we're talking about functions now.

Reflective:

It's kind of fun to see them give a theorem using functions and for the proof, just ask you to adapt the proof from chapter 2, using functions in place of integers.

9.4, Due February 11

Difficult:

I was a little confused in the proof of Lemma 9.26 with all the multiplication that was going on. They didn't explain WHY they were doing anything, so it was harder for me to make the logical connections and understand the logic behind the proof.

Reflective:

I mostly just thought it was cool to see these equivalence relations. I hadn't ever really thought about why 1/2 =2/4 and so on and so forth. This is the kind of thing students in junior high and high school get confused about, and as a math ed major, it's cool to see an explanation.

Questions, Due February 9

Which topics and theorems do you think are the most important out of those we have studied?

Obviously any theorem with a name, such as "The Remainder Theorem", "The Division Algorithm" and so on and so forth. It seems like these first 4 chapters have been about division, factors, addition/multiplication operations

What kinds of questions do you expect to see on the exam?

I'd expect some questions about definitions (such as, "What is a ring?"), questions that apply definitions ("give an example of a ring homomorphism"), and questions about theorems (stating, applying, and proving).

What do you need to work on understanding better before the exam?

Mainly I need to work on understanding chapter 4, especially getting a better understanding of irreducible polynomials and how to find them. I also have a hard time with "surjective", more specifically with how to prove that something is surjective.

4.4, Due February 7

Difficult:

I am still struggling with the idea of irreducibility. The homework due today asked us to list some irreducible polynomials. The answers were in the back of the book, but I still didn't understand exactly why it was irreducible. The Theorem 4.18 helps me to understand a little bit better of how to determine which polynomials are irreducible.

Reflective:

I find the remainder theorem to be really cool :) I hadn't realized that if we take the divisor of the form (x-a) then the remainder will be f(a). I'm wondering now, can we always get a divisor of that form? What happens when we don't?

4.3, Due February 4, 2011

Difficult:

I found the concept of an "associate" very hard to understand. The book didn't really provide a good example that could solidify in my mind what it means. If I understand correctly, if a=3 and b=1/2 then u would be 6 so that a=bu? It's much more difficult to think about in terms of polynomials....

Reflective:

The idea of a polynomial being "irreducible" is interesting. It's technically a way of finding "prime" polynomials, but for some reason they thought "irreducible" would be a better term :)

4.2, Due February 2

Difficult:

I think I am still struggling with what a 'field' is in context of the book, so I get a bit confused when it talked about fields in regards to some of the theorems in this section. I'm also having trouble understanding the GCD in regards to a function. The theorem made sense to me right up until they gave the example of the GCD of 3x^2+x+6 and O. I don't really understand why they multiplied it by 1/3.

Reflective:

As the book mentioned, most of the results from divisibility and greatest common divisor are almost the same as they were for regular arithmetic. The necessary modifications to apply the theorems to F[x] were intuitive, for the most part.

4.1, Due January 31

Difficult:

When I read Theorem 4.2, I was pretty confused at first. I'm not sure what I thought it was referring to, but I already know that x^4*x^2=x^6 because you add exponents when you're multiplying them together. I don't think I've ever heard them referred to as "degrees" before, so when I read deg[f(x)g(x)] = deg[f(x)]+deg[g(x)] I was thinking more of regular arithmetic than exponents.

Reflective:

The Division Algorithm In F(x) seems almost exactly the same as the regular division algorithm, but they don't require r(x) to be greater than zero. I didn't totally follow the proof, and perhaps it was addressed in there, but I'm curious as to why r(x) doesn't have to be greater than zero...

Questions, Due January 28

How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?

I probably have to spend as much as 2 hours per assignment and even then it's not necessarily done. That's just how proofs are for me, they don't come easily. I feel like the lecture and reading are basically the same. I generally don't get much more from the lecture besides what I already got from the reading.

What has contributed most to your learning in this class thus far?

The interaction with others (outside of class) - working with peers on homework or one-on-one attention from Dr. Jenkins.

What do you think would help you learn more effectively or make the class better for you?

I think I need to continually review past chapters and their theorems to help me remember which ones to use. I also think it would be valuable for Dr. Jenkins to go over a few homework problems at the beginning of class. If time is a concern, we can skip some proofs, as we already read them in the book, and I don't think all proofs need to be rehashed in class.

3.3, Due January 26

Difficult:

I'm not sure if I understand why we even care about rings being isomorphic. It was strange to see the + and x tables of Z10 written "out of order" from what we would originally do, but I could understand why they chose to do it that way. However, what do we get from 2 things being isomorphic? Why does it matter that Z5 and Z10 are isomorphic?

Reflective:

This particular section brought up a few mathematical ideas that I haven't seen for a while, namely surjection, bijection, and image. I only vaguely remembered those concepts, so I ended up having to go look those definitions up again!

3.2, Due January 24

Difficult:

I am still thinking of operations in Z when I'm looking at some operations on rings. I was watching the example on pg 59 where they have (a+b)^2 and all seemed well and good until they mentioned that we can't necessarily say "ab+ba=2ab" because ab might not equal ba.... I have to keep reminding myself that in Z6 it's ok to have 1-2=5.

Reflective:

I found it interesting that this book uses the term "cancellation" but only in light of what has been talked about in my practicum class (as I'm a math ed major). We had talked about how "cancelling" gives the wrong message, like we can just arbitrarily take out or add in what we need to, when really we are "simplifying" not "cancelling."

3.1, Due January 21

Difficult:

It was hard to follow the example of the Cartesian product Z6xZ. I was confused when they said that a,a' are both in Z6 and z,z' are both in Z. It seemed like it would make more sense to have a,z in Z6 and a',z' in Z. It wasn't until reading in Appendix B and reading theorem 3.1 that it actually made sense to me.

Reflective:

In class I think I was a little confused about subsets of set that are rings, but I think now I understand. While Z is a ring, and E is a subset which also happens to be a ring, O is not a ring. It's ok for subsets to not have all the properties of a set. I think I had always thought of it the other way around.

2.3, Due January 14

Difficult:

I was able to follow the theorems and their proofs, but when they applied it in the example problem I realized I didn't really have a grasp of the usefulness of the concept. After reading the example of 24x=5 in Z95, I had to go back and re-read the theorems again to understand what they had done to arrive at their answer of x=20.

Reflective:

I definitely appreciated the footnote directing me to Appendix A so I could remind myself what it means when they say "the following conditions are equivalent" as I had forgotten what exactly that meant. I didn't realize how much reading the various Appendixes would help me review my basic logic and proof skills!

2.2, Due January 12

Difficult:

The most difficult part to wrap my mind around is the way ordinary arithmetic might look when using modulos. It may be true that in Z(mod 5), 4+1=0 or 3x4=2 or 4+4=3 - usually when we see an equation set up like that, we intuitively assume that they are just regular integers and the equations just feel wrong!

Reflective:

It was definitely easy to recognize all the basic operations that could be performed on Z(mod n). It was also easy to follow the proofs the book presented for valid operations in Z(mod n). I found this chapter much easier to connect to previous chapters and theorems than any other section.

2.1, Due January 10

Difficult:

I found it difficult to remember what "a=b(mod n)" means. As they were going through various theorems and proofs, I had to keep going back to the definition of congruent modulos. Even more difficult was the concept of the congruence classes of (mod n) and it's accompanying theorems and proofs. I will likely need to re-read section 2.1.

Reflective:

I did find it curious that when we talk about 2 sets of congruence classes, it's only a matter of disjoint vs. identical. Usually 2 sets can have some elements in common but not be the same sets, but with congruence class sets, that can't happen.

1.1-1.3, Due January 7

Difficult:

I had a hard time following the back substitution portion of the Euclidean Algorithm. I think I understand the reasoning and process when they apply the division algorithm to find (324, 148), but I don't understand why or how they applied back substitution. If I understand the algorithm, 4 is the GCD (324,148)? Then why the back substitution?

Reflective:

In my Math history class that I'm taking through independent study, I need to prove that the square root of a prime number is irrational. I've been struggling to understand or create a proof for such. Now that I've learned more about the division algorithm and it's application, as well as learning more about primes, I think I will have a much easier time creating the proof!

Introduction, due on January 7

What is your year in school and major?

I'm a MathEd major in my senior year here at BYU.

Which post-calculus math courses have you taken?

I've taken Math 190, 214, 315, 334, 343, and 362.

Why are you taking this class?

Well, it's required by my major. Honestly, I would never have picked it myself. I know it's going to be a lot of work for me - I struggle with courses that rely heavily on proofs. Proofs have just never come easily to me.

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 my favorite teacher was my Pre-Cal teacher in high school. It was obvious that she was enthusiastic about the subject and that she cared whether or not the students were learning. She tried very hard to relate math to the students - often through movie clips! It was an enjoyable class :)

Write something interesting or unique about yourself.

I've been married for 5 years and I have 2 kids. Benjamin is 3.5 and probably the smartest kid you'll meet. Tristan is 17 months and a little terror. By the end of the month I'll be finishing my basement - by the end of the year, I'll be graduating - and sometime in the next 5 years I will become the owner of a tax practice that supports about 600-700 clients currently. It's going to be a crazy next few years!

If you are unable to come to my scheduled office hours, what times would work for you?

The perfect time for me would be MWF 10-11 (just before 371 starts).