Last Blog Entry! Due April 11

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

 I think Legrange's Theorem, the remainder theorem, and the first Isomorphism theorem are the most important theorems we have studied. Some of the topics I think are important are the properties of groups and how the groups and rings are isomorphic to other groups and rings.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class.


 I think one of the hardest things to do is finding a function to create an isomorphism between two groups or rings. One example of this is problem 15 on the homework review. Another concept I am struggling with is proving that a group is cyclic, like problem 8, 14, and 18. I would really like to go over problem 1 in class though, with An being a normal subgroup, and also problem 12 would be good since I am don't have the best understanding of automorphisms.

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

Throughout this semester, I have realized that there are things in our every day life that we may be able to consider a group, and since we know so many things about groups now, it seems like I could understand those everyday things better, like the rubiks cube, or the equivalence classes when painting the sides of a cube (i am in combinatorics and we discussed this idea).

Section 9.4, Due April 9

Part I: What was the most difficult part of the material for you?

I don't really understand why [a,b] + [c,d] = [ad+bc, bd] in this section because in my mind, I would think it should be [a,b] + [c,d] + [a+c, b+d] but it is not even close to that. So why not? 

Part II: Write something reflective about the reading.

I see how the field F and the integral domain R relate to the rationals and the reals, but I am wondering how any of this stuff came up. Why would someone want to find something that is isomorphic to the rationals and reals when we have them already? I know I probably can't get an answer unless I studied more math, but it is something I am thinking about. 

Section 8.4 & 8.5, Due April 6

Part I: What was the most difficult part of the material for you?


I am stuck on theorem 8.21 because I am having a hard time seeing how the number of distinct conjugates of a are related to [G:C(a)]. The proof really confused me because I don't really know why they are doing what they are doing and what it is accomplishing. The example kind of makes sense, but the proof is way to confusing for me to see that clear connection.



Part II: Write something reflective about the reading.

I thought it was interesting to learn about conjugates and conjugacy classes. At first, I didn't like thinking of another kind of class, but I can see how this class is similar to congruence classes with modular arithmetic, it is just a computation and a set of elements with that computation. At least, that is how I kind of think about it. 

Section 8.2, Due April 4

Part I: What was the most difficult part of the material for you?

I was most confused once Theorem 8.15 was introduced. I don't really understand the x^-1Kx and why that is needed for this section. I think it would be more helpful with an example because most of the other theorems have examples but theorem 8.15 doesn't have any, so I am confused. 

Part II: Write something reflective about the reading.

Most of this chapter I have been really confused, but the First Sylow Theorem actually made sense and seemed like an interesting extension idea with groups. So I am glad that this section is making a little more sense. 

Section 8.1, Due April 2

Part I: What was the most difficult part of the material for you?


There was a lot of stuff in this section, but one of the most confusing parts was the definition of invariant factors and elementary divisors of G. The book gave an example but I don't really understand it still and maybe it is because I don't really understand the previous theorems but I was very lost at this part.



Part II: Write something reflective about the reading. 


So I am really struggling with chapter 8, I just don't see what we are trying to do and where this is going. Chapter 7 made some sense because we were relating many of the things from rings to groups, but now it just seems like random ideas that don't make any sense to me and I don't understand the importance of it all, but maybe it will change as I learn more about this section.

Section 8.1, Due March 30

Part I: What was the most difficult part of the material for you?

I don't understand the definition of the Cartesian product of G1, G2, ..., Gk because the book says that it is (a1, a2, ..., an)(b1, b2, ...bn)=(a1b1, a2b2, a3b3, ...,anbn). That doesn't make sense to me at all, because I think of it as G1=a1,a2,a3,....,an or G2=b1,b2,....,bn). So I don't see where G3, G4,....,Gk came into the product.  

Part II: Write something reflective about the reading. 

The first example in the book is really interesting since the books shows us the connection that M and N make to MxN. I don't really understand why all of this may be useful to us working in groups, but it seems interesting. It feels like we are just finding ways that other things about groups are similar to simpler groups, so I guess I just wonder why we even study the complicated stuff when it is similar to the simpler ideas in group theory.

Section 7.10, Due March 28

Part I: What was the most difficult part of the material for you?


Because we haven't really discussed the alternate group yet, I think the whole proof of 7.52 was really confusing. I am still trying to understand multiplying a permutation with cyclic notation and I am struggling the proofs involving cyclic notation, so this section didn't really help with that since a lot of it was about that stuff.



Part II: Write something reflective about the reading.


I think Lemma 7.53 is really interesting because I understand it well and the proof makes a little more sense than the rest, but it seems like a really good way to understand the structure of alternate groups. Plus, it is crazy that this section was pretty much written so to prepare the reader for a proof so this lemma seems pretty important.

Section 7.9, due March 26

Part I: What was the most difficult part of the material for you?

I was very confused with the transposition part. I do not understand corollary 7.48 because I don't see at all how (1234)=(14)(13)(12). So after I read that part, the rest of the section was very confusing.

Part II: Write something reflective about the reading.

Even though I didn't understand the transposition idea, I understood the disjoint cycle notation because I am in Math 450 right now and we have discussed this subject before. It is really interesting taking these classes together because they have a lot of parts that are similar so it is easy for me to make connections.

Section 7.8, Due March 23

Part I: What was the most difficult part of the material for you?

The third part of Theorem 7.44 confused me the most. I didn't really understand the proof and where T came from. I don't really see how that part of the theorem is even helpful, and so I guess I don't know what it is meaning in "English."

Part II: Write something reflective about the reading.


I am taking Math History right now and we are discussing set theory and group theory and how that started to develop, and reading this chapter made me think back to the different views on math in the 20th century. I always thought math has always been there and we discover it, but reading this chapter gets me thinking of the other view that math may have been something we created to solve our problems. I can see how group arithmetic is starting to be similar to normal arithmetic and it seems like it was sort of constructed so it would be like that. Crazy.

Section 7.7, Due March 21

Part I: What was the most difficult part of the material for you?

 I think I am still struggling with a specific example of a quotient group, but I think after the lecture tomorrow and after the homework, it will make more sense to me because I will be able to actually work with it.

Part II: Write something reflective about the reading.  


One of the examples was with the integers and the rational numbers and how Q/Z is a quotient groups but no a quotient ring. That is crazy because they try and relate much of what we are learning with groups back to rings yet even though something may be group, I guess it is possible that it won't be a ring.

Rest of Section 7.6, Due March 19

Part I: What was the most difficult part of the material for you?


I had to use Theorem 7.34 in the homework, but since I hadn't read about it at that point and we hadn't discussed it as a class, I was a little lost on that. I kind of make sense of the proof but it is a pretty big theorem with 5 parts all equivalent so I think I just need to have a better understanding of that proof.


Part II: Write something reflective about the reading.  

I really like how this section led up to theorem 7.33. I feel like what we did in class and what the book describes eventually gets you to the point that theorem 7.33 is true, and then they state it and it just fits perfectly.  

Section 7.6, Due March 16

Part I: What was the most difficult part of the material for you?

 The very first example in the book is what started my confusion. It isn't a big confusion but I am still confused. I don't understand how r1*t^-1=r1*t. I really just don't understand what the inverse of t is and such. For the most part, this section was easy to understand, but I am a little lost on the group D4, so I think I just need to revisit that. I also was confused on how theorem 6.5 for groups would only work on Abelian groups, but I am sure we will discuss that more in class.

Part II: Write something reflective about the reading.  
 
 It is really interesting to connect the ideas of rings and ideal to groups now. It scares me a little bit because ring quotients were pretty confusing so I hope that connecting those ideas to groups isn't too confusing.
 

Rest of Section 7.5, Due March 14

Part I: What was the most difficult part of the material for you?

 I am still struggling with theorem 7.28. I used it in the homework already but I think that confused me more because I don't really grasp what the theorem really means in "English."
 
Part II: Write something reflective about the reading.  


I think it is incredible that some of the groups of certain orders are isomorphic to a certain amount of groups. I remember when studying for the last test, the TA mentioned this idea and it blew my mind away. At first when learning about groups, it seems like there are so many possible ways, but now it seems like the list can be narrowed a little bit which is cool.

Section 7.5, Due March 12

Part I: What was the most difficult part of the material for you?

 I am struggling with the proof of theorem 7.25. First, I am just confused on why it is important, I guess it is hard for me to understand in "English" and the proof for the first part I'm not quite understanding. I think what is the hardest for me to grasp is the meaning of a right coset.

Part II: Write something reflective about the reading.  


I loved the rules that they gave at the beginning of how multiplication and addition relate. It made sense and I understood why in groups, we have to distinguish the two yet we can see how they still both work.

Section 7.4, Due March 9

Part I: What was the most difficult part of the material for you?

The proof for Cayley's Theorem had me really confused. I don't really understand what A(G) really is and it is hard to follow the proof without understanding that notation. It seems like such an abstract proof, so I hope I can understand it in class.

Part II: Write something reflective about the reading.  

From working on the review for the exam and looking at all the different groups and isomorphisms from rings, it doesn't surprise me that we can relate groups and rings with isomorphism and that groups can be linked to the permutations. I really don't know how anyone was genius enough to come up with all this stuff!  

Questions, Due March 7

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


I think some of the most important topics is understanding F[x]/(p(x)) and what that means. Also, the first isomorphism theorem is quite important along with the proof. Examples and non-examples I think will also help a lot.

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


I expect to see questions that ask to list examples and non-examples and various things. I also expect to see questions that deal with F[x]/(p(x)) and possibly listing the possible congruence classes. Also, proofs with groups and if something is a group.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I need to work on thinking of examples of maximal ideals, like with problem number 5 as a sample problem it says, Give an example of a maximal ideal in a ring that does not contain all proper ideals of the ring.  I also need more work on describing what a quotient ring means. So like problem 6 says to give an example of a prime ideal in ZxZ that is not maximal and describe the quotient ring ZxZ/I.

Section 7.3, Due March 3

Part I: What was the most difficult part of the material for you?

The proof for theorem 7.15 was really confusing. It kept referencing back to previous theorems and I kept trying to make sense of the flow of the proof, but I didn't understand how they got that every element of G is a root of the polynomial x^m -1. I don't even understand what that means and why it is even mentioned in this theorem.

Part II: Write something reflective about the reading.  

At first the cyclic subgroup generated by a made sense and it seemed pretty easy to grasp. Then when they talked about a subgroup generated by S, it seemed a little bit difficult but then I saw the relation between the two and it made more sense.  

Section 7.2, Due March 1

Part I: What was the most difficult part of the material for you?

Theorem 7.8 was a lot to take in, since it really is like 4 theorems. When reading the proof for 3 and 4, I was really confused and I think it will help to actually go over them in class. The proof for part 2 was a little confusing as well, but it makes sense why proving 2 would also prove part 1. 

Part II: Write something reflective about the reading.  

The finite order of an element a in a group is a really interesting idea. I first tried to think of examples to make sense of what this means, but in the integers and rational and real numbers, there is no number that has finite order. Then, when I read the examples in the book, it made sense of what kinds of elements would have finite order.  

Rest of Section 7.1, Due February 29

Part I: What was the most difficult part of the material for you?

I can understand the part of how a ring is an abelian group under addition, but I think I am still confused on how it doesn't work with multiplication. And since a ring is an abelian group under addition, does that mean that the binary operation is addition and you only need that operation. I guess I am still confused on what a binary operation still means. 

Part II: Write something reflective about the reading. 
 
When they book started taking about a cut out square and rotating it and reflecting it, I thought of the example in class with the Rubik's cube. It is really cool to make that kind of connection and to see how both are groups.

Section 7.1 first half, Due February 27

Part I: What was the most difficult part of the material for you?

The most difficult part of the material is that a group has a binary operation *. At first I didn't think much of what that was saying, but then when they went into the examples of compositions of functions, I realized that the composition was the binary operation, which seems weird. 

Part II: Write something reflective about the reading. 

I am in a combinatorics class right now and we are talking about permutations, so I could easily see the connection of why the order of S(n) is n! and that the notation for these functions. It was really cool to make the connections of this section to my combinatorics class.  

Section 6.3, Due Februray 24

Part I: What was the most difficult part of the material for you?

The section about maximal ideals really confused me. After thinking about it for a while, the example given with (3) kind of made sense how (3) is a maximal ideal, but then the example of Z[x]/(x) really lost me and I was just lost on what that even meant. 



Part II: Write something reflective about the reading.

It is becoming really difficult to tie everything from what I have learned in previous sections to this chapter. I am still struggling with the idea the Z/(3) is the same as Z mod 3 and I just have a really hard time trying to figure out the cosets of R/I for some example. It takes a lot of reflection to really understand this.   

Rest of Section 6.2, Due February 22

Part I: What was the most difficult part of the material for you?

 I am most confused about the proof in theorem 6.11. The second part of the proof is assuming that f is injective where f:R to S and f is a homomorphism of rings with kernal K, and we are trying to prove that K = (0subR). I can understand most of it until they say that f(c)=0sub s and that f(c)=f(0subR). I do not know why f(c)=f(0subR). I kept trying to look at how the kernel fits into this equation and I am just getting really lost at this point.


Part II: Write something reflective about the reading.  


This section is really becoming more and more abstract. I am having such a hard time with making a connection with some of this stuff to things we have already done, but maybe there are no connections to really make yet from past things I have done. I just hope that working on the homework and seeing other example will help me understand what some of these theorems are even saying.

Section 6.1-6.2, Due February 21

Part I: What was the most difficult part of the material for you?

I am most confused about the set R/I. I feel like I understand what an Ideal is and how it is related to R, but now connecting it to modulo of R seems too confusing and abstract. I don't think I am comfortable yet with cosets and that is probably where most of this confusion is coming from.


Part II: Write something reflective about the reading. 


The notation in this section starts to get really confusing. Since a+I denotes a congruence class yet they then define addition in a similar way and it is strange. I don't really understand why they didn't use a different notation for addition or for the congruence class, like something that related better to Chapter 2.

Section 6.1, Due February 17

Part I: What was the most difficult part of the material for you?

I was most confused on the example given for what an ideal isn't. While the concept and term of ideal seems to be understandable, the book didn't seem to give a concrete simple example of a principal ideal generate by some number.


Part II: Write something reflective about the reading. 

While I have never heard of the math term ideal before, the book seemed really clear about what it was and it made a lot of sense with the example they use at the beginning of the chapter. Normally I don't understand new math terms very well at first, but this one was quite easy to grasp.

Section 5.3, Due February 15

Part I: What was the most difficult part of the material for you?

 I am most confused about the definition of an extension field and a subfield. With how they used subfield, I felt like it should have been used the other way, like K is a subfield of F. And so that is why I am even more confused with the extension field definition. It doesn't make sense to me with how it has been for subrings.

Part II: Write something reflective about the reading.

  At first I thought it would be too obvious to say that F[x]/p(x) is a field if p(x) is irreducible because F[x] comes from a field, but then I realized I was generalizing too much. Even though F is field, I guess it is still possible that F[x]/p(x) is not a field when p(x) is reducible, which is really crazy.

Section 5.2, Due February 13

Part I: What was the most difficult part of the material for you?


I was most confused about Theorem 5.7. The second sentence was way too confusing because it used a symbol for mod and then had the word modulo and I didn't understood how those related to each other, and I still don't. Maybe the book just misprint it incorrectly or I am just not understanding the notation correctly, but I don't understand what the theorem is saying in English. 


Part II: Write something reflective about the reading.


It wasn't until I started reading this section when I realized that F[x]/p(x) is similar to the idea of integers mod n. Once I understood that notation in that way, it started to click better. When I first saw it, just knew what you would say if you saw that notation, but I didn't really understand what I was saying. Polynomials are a lot harder than integers to understand for me. 

Section 5.1, Due February 10

Part I: What was the most difficult part of the material for you?


I understand what congruence class is, but I am so use to it in the integers mod that I am a little confused about how it is in a polynomial ring. Even understanding mod in a polynomial ring just seems so much more complicated.



Part II: Write something reflective about the reading.

It is really cool to see how each section of previous chapters relates to a different structure of "numbers" and I can clearly see that in this section. Many of the theorems are worded very similarly and it is helpful to look back at the previous sections to make those connections. 

Section 4.5-4.6, Due February 8

Part I: What was the most difficult part of the material for you?


I am lost at the start of theorem 4.20 (Rational root test) because it discusses about the variables r and s, but I don't even understand where they got an s! I know that is comes from the root of f(x) but I guess it is worded strangely and I would need an example that I work on to see the connection.



Part II: Write something reflective about the reading. 

Theorem 4.25 I think is really amazing because it seems impossible that EVERY non-constant polynomial in the C[x] has a root in C but after seeing the proof and thinking about it, it makes sense. But still, it is surprising.  Also,  many of the theorems after had similar ideas with finding things dealing with EVERY non-constant polynomials in a polynomial ring, which is really amazing.

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

The Euclidean Algorithm, the fact that we can write the gcd of f and g as a linear combination for some u and v, and the theorem that says "If F is a field, then for any f in that field polynomial and any a in F, the element a is a root of f iff (x-a) divides f." 

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

I expect there to be proofs, definitions, and many some small calculations that we may have done similar in the homework. But I think the proofs will be proven with the theorems and definitions we have memorized. 

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I am still trying to understand better how to prove if a function is surjective or not. Most of the time it just seems obvious, so I don't know if I am missing something or if I understand it so well that I am making it more complicated.  One example is, problem 15 on 3.3 which says, Show that the complex conjugation function f:C to C (whose rule is f(a+bi)=a-bi) is a bijection.

Section 4.4, Due Feb. 6

Part I: What was the most difficult part of the material for you?

 The most difficult part was differentiating between what x is as an indeterminate or a variable. I am just confused at what the phrase, "the function f is induced by a polynomial," which makes it hard to understand what an indeterminate is and what a variable is. I feel like I may be getting confused over something that is simple, so maybe it isn't as hard as I think it is.

Part II: Write something reflective about the reading. 

The Remainder Theorem is an interesting theorem that I am kind of shocked is a theorem. Yet when I look at the proof, it makes sense. I guess I need to try a few problems to believe it better for myself.

Section 4.3, Due February 3

Part I: What was the most difficult part of the material for you?

The definition of an associate is when I started to get a little lost. I think it would have made more sense to me if the book had given examples that showed the idea. Then when I read the definition of irreducible, I was really lost because I had to understand what an associate is. The example afterwards helped me a little bit, but I am still most confused about what associate and irreducibility is.

Part II: Write something reflective about the reading.

I understand that his section connects to a previous section we did with the integers and prime, but the book didn't really focus so much on connecting the idea of irreducibility in with primes. I eventually saw some of the equivalences on my own which was interesting to make the connection myself. It would have been more helpful though I think if the book did connect these two ideas more so.

Section 4.2, Due Wednesday Feb. 1

Part I: What was the most difficult part of the material for you?

The proof to Theorem 4.5 was the hardest thing for me to follow. It is difficult to read and follow the proof all the way through to show that d(x) is the unique gcd. I understand how there is a gcd but it seems more difficult to see that it is the only one.

Part II: Write something reflecting about the reading.

It is interesting to compare these theorems to chapter 1 and see the similarities. I never would have expected polynomials to act so similar to integers. I am just guessing that the next section will continue on into showing how mod works with polynomials as well. 
 

Section 4.1, Due on January 30

Part I: What was the most difficult part of the material for you?

I am most confused on what the F[x] represents and if f(x), g(x) is an element of F[x]. Does that mean g(x) and f(x) are the coefficients for the polynomial or does it mean f(x) and g(x) are the coefficients with the x part, so the whole term? This is what was mentioned in Theorem 4.4 The Division Algorithm in F[x] and I just don't fully understand the concept of F[x] yet.

Part II: Write something reflective about the reading.

 I really like seeing how the division algorithm works and the proof they use in the book. It was really helpful to see an example next to the proof so you could understand what they were meaning better. Even thought I was confused with the meaning of Theorem 4.4, I think I can understand the computational side of how to do it which I like.

Other Questions, Due January 27

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

I spend at least 2 hours on each homework assignment. I try to work with other classmates whenever I can and I go to the T.A.'s office hours so I can understand it the best. The lectures and the readings help me prepare for the homework, but there are still some questions on the homework I may be completely lost on even though I did the reading and went to class.

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

Working with groups on the homework has helped me the most. But I also think that reading the section before the lecture has been helpful once I go to class because most of my questions will be answered.

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

There are times when I have worked hard on the homework and I still have a few questions, so if time allows for it, it would be nice to go over a question on the homework at the  beginning of class.

Section 3.3, Due on Januray 25

Part I: What was the most difficult part of the material to you?

 The most difficult part of the reading for me was the statement and proof for Corollary 3.13. I am really confused about the idea of the image and how it relates to S. I understand that the image is f(r) for any r that is from R, but it is hard for to me relate this idea in the proof since I is just stated to be the image, when I could say that I want to see f(r) to be the subset of S, right?

Part II: Write something reflective about the reading. 

I really liked the example in the book about how to find a function that is an isomorphism and it had integers mod 12 and the cross product of integers mod 3 and 4. It built up so well to each step and to the final product, yet it seems super complex because there are so many subscripts and numbers with functions, cross products, and integers. It could be easy to get lost but you just have to look at each step carefully and it makes sense.

Section 3.2, Due on January 23

Part I: What was the most difficult part of the material to you?

The most difficult part was understanding the proof with theorem 3.6. It was hard for me too see the connections from this theorem to Theorem 3.2 and I think if the book had made it more organized rather than used numbers to relate back to statements, I would understand it better. Hopefully in class it will make more sense as to how only those 2 statements in Theorem 3.6 are needed to prove a subring rather that the 4 statements in Theorem 3.2.

Part II: Write something reflective about the reading.

I really like the definition at the end of this section about what a zero divisor is because the real numbers and the integers do not have a zero divisor but the integers mod six have one  and so many others. For most of this section, it kept showing how the integers and rings have so many common properties, but at the end they were different properties. Connecting these properties from the integers into other rings seem too easy at first because these properties are so second nature, but the zero divisor is easier to see the importance because it isn't in the integer.

Section 3.1, Due on January 20

Part I: What was the most difficult part of the material for you?

The first example that was mentioned was confusing for me to follow just because they mentioned that the product of the integers mod 6 crossed with the integers is defined in Appendix B but I couldn't find it. I eventually moved on from finding what it meant and I still understood the problem but I didn't understand with what it meant by integers mod 6 crossed with the integers.

Part II: Write something reflective about the reading.

I think it is interesting that the axioms needed to prove a subring are so much shorter than to prove if something is a ring. I feel like there is more to think about in this theorem so that I can make more sense in why we must only prove those 4 axioms, but I am willing to believe because of the examples that were done for now. But with the fourth axioms for a subring, I need to think more about that because it seems like we should find if solutions exist but apparently we already know they exist.

Section 3.1 through middle of page 48, due on January 18

Part I: What was the most difficult part of the material for you?

The most difficult part of the material for me is trying to match up all the definitions with certain groups of numbers. Once the definition of a field came up, I figured that, for example, since all real numbers are a field, that would mean they satisfy axioms 1-12 but then I noticed that the definition of a field doesn't mention Axiom 11 but apparently it does hold in fields. That is when I get a little confused because it seems that Axiom 11 should be mentioned then.

Part II: Write something reflective about the reading.

I really liked the example on page 43 where they show a set T={r,s,t,z} and that they define it in a way that it is considered a ring. It is so crazy but yet it makes sense because it follows the definition. Since Axiom 9 and Axiom 10 were mention right after the definition of a ring that there aren't many examples of just rings without Axiom 9 and 10, but this example at least showed a ring that isn't even a commutative ring, and I noticed that before I read that part of the reading so I was making sense of everything pretty well.

2.2, due on January 11

Part I: What was the most difficult part of the material for you?

     The most difficult part was when they used an example to show that addition and multiplication works with those congruence classes in the integers modulo n. This was on the first page of the section and I wanted them to explain it more. I feel like there is a lot of ambiguity. If they broke it down from congruence  classes, then to integers, and back into the classes, I would believe the example.

Part II: Write something reflective about the reading.

     I think it is really cool that many of the properties that work in the integers carry over in the integers modulo n. I feel like a whole new world has been opened up and there is so many new things to discover about the integers modulo n. It will seem difficult though to follow the changes the  books make in representing the congruence classes and to remind myself that they are classes, not integers. 

2.1, due on January 9

Part I : What was the most difficult part of the material for you?

       The most difficult part of the material for me is the connection between Corollary 2.5 to Theorem 2.3 and to the definition of congruence class. I can connect the definition of congruent to modulo and the remainder idea, but I am having a lot of difficulty connecting everything together. I think what makes it so difficult is to use so many variables rather than actual numbers to see the pattern. The congruence class is an abstract idea for me right now because I haven't written it out much yet, but hopefully as I work through the homework and look through our notes in the future I will get use to it.

Part II: Write something reflective about the reading.

      This section reminds me of learning about modulo in Math 290. That is probably why I can connect some of the pieces together but since it has been a little while, I am getting a little confused and lost. In Math 290 I learned more about the connection of how modulo relates to the remainder in the equation a=qn+r where r is the remainder. Yet I didn't go in depth about modulo arithmetic in 290 so hopefully I will get better with it in this class.

1.1-1.3, due on January 6

Part I : What was the most difficult part of the material for you?


     The end of 1.2 was the most difficult part of the material for me. I was definitely getting lost in all the equations and the implications and theories, but at the end of 1.2 I was confused about Theroem 1.6 The Euclidean Algorithm. I am still a bit confused on it. I understood it well in class but reading the actual statement of the theorem doesn't make sense to me with how the book writes it out. I am mostly confused with the second sentence which states, "If b|a, then (a,b)=b." I am confused because I can think of an example such as 3|27 but (3,27)=9 not 3. This also satisfies the first statement which is that a and b are positive integers with a>b or a=b, and in my example 27 > 3. So I just don't understand that statement.


Part II: Write something reflective about the reading.

     I think the most interesting part is the proof for Theorem 1.11 The Fundamental Theorem of Arithmetic. I liked it because it looks long and complicated but it makes complete sense. I also feel like it will lead into a lot of important proofs in the future so it seems like this is an important theorem and proof to remember and memorize. This whole section of reading goes into the idea of number theory which really fascinates me, but I guess the ending just had a good punch to it.  

Introduction, due on January 6

• What is your year in school and major? I am a junior at BYU majoring in Math Education.
• Which post-calculus math courses have you taken? I have taken Math 290, Math 313, Math 314, Math 334, Math 341, and Math 362
• Why are you taking this class? For the most part, I am taking this class because it is required for my major. I don't really know what abstract algebra is about but I do hope to gain a deeper understanding of the mathematics that will be most helpful for me in this subject to help me teach junior high and high school students better. I believe that this class involves a lot of proofs and I hope that I can make sense of the proofs and improve on writing proofs.
• 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? One professor I had that was most effective was one that answered questions that the students had and that was patient whenever I would ask for help in his office hours. I always do better when I can visit with the professor and discuss the math with him. I also think that the simpler and the clearer a subject is taught in, the better I understand the concept.
• Write something interesting or unique about yourself. I really like the show Buffy the Vampire Slayer and I like to do puzzles. My best days are when I do them both at the same time.
• If you are unable to come to my scheduled office hours or the TA's scheduled office hours, what times would work for you? I am not able to come to your office hours, but I can make it to some of your TA's office hours time. I would prefer to still talk to you at some point and the best times for me are 12:00-12:50 MWF and 3:00-3:50 MWF or any time on T, Th.