Review, due on April 13

Which topics and theorems do you think are important out of those we have studied? - [All the ones about rings, groups, and ideals.]
What do you need to work on understanding better before the exam? - [Probably a little bit of everything]. Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Wednesday. - [I would like to go through as many examples as possible listed in the section entitled, "Know lots of examples of all the things we talked about, especially:..."]
How do you think the things you learned in this course might be useful to you in the future? - [I am going to be taking Number Theory, so I figure this course would be very useful in understanding Number Theory. Other than that, this class has really taught me that I can understand difficult material and overcome difficult challenges.]

8.3, due on April 11

1. I want to know where they got the name Slyow! Also, in the example on page 8.3 says that S6 says there will be at least 60 subgroups of order 4. How are you supposed to figure that out? Is there a way to figure that out? We might have talked about how, but I am just not recalling it... Also, what is the relationship between abelian and normal? They seem pretty similar...

2. I don't know what else to say about this section, except that it is 8.3 which means we are almost done with the course which means I won't have to do these blog posts any more and I am so excited about that!!! :)

8.2, due on April 8

1. I feel like this section has a lot of interesting information to offer, but I didn't understand hardly any of it. Yep. There was a lot of it. I probably should have given it a little more time than I did to digest it all because there was so much of it.

2. I liked Theorem 8.12. It just made sense to me that if G and H were isomorphic to each other, then they would have the same elementary factors, or prime powers. The theorem was a bit to digest however, but it makes sense for the most part.

8.1, due on April 6

1. I think the only tricky part about this section was the proof of theorem 8.1. I guess that wasn't even too difficult because I am feeling like I am grasping how to prove isomorphisms a lot better now that we have done the First Isomorphism Theorem a few times.

2. This section was relatively simple. I especially thought Lemma 8.2 was cool. I did not see the significance or newness it brought until I realized we were talking about ab=ba where a and b are elements within DIFFERENT subgroups! That is crazy! Before we only talked about elements being normal within a single group.

7.10, due on April 4

1. I think I need to do my homework for section 7.9 because I don't think I am following too well this new notation for permutations. Like, I am pretty lost in this huge proof. I guess that is how most chapters end - hardest stuff at the end.

2. It was cool to see stuff about normality in this section, even if I didn't understand the permutation stuff. I feel like I understand normality pretty well.

7.9, due on April Fools!

1. I don't think I'd be able to prove Theorem 7.46 or 7.49. Also when they started doing the 2-cycle transposition, such as (1234) = (14)(13)(12), I can't follow it as easily as say, (243)(1243). I hope we'll go over transpositions in class.

2. THIS SECTION WAS SO MUCH EASIER than the past few! I love the new notation for permutation, it is much more economic and interesting - it makes you think a little more. I also think it is interesting how the First Isomorphism Theorem keeps on coming up in the sections.

7.8, due on March 30

1. Did we ever do a THIRD ISOMORPHISM THEOREM FOR IDEALS? Because I know we did a FIRST one for ideals, so 7.42 does not look very new - 7.43 does however. This section is full of pretty intense proofs that I am not at all positive I would know how to prove on my own. Most likely I will have to for the test though!

2. I liked Theorem 7.45. It deals with the integers(mod p) where p is a prime. I think primes have a lot of very unique and interesting applications that have shown up a lot over the course of this course.

7.7, due on March 28

1. I know I have mentioned this before, but I still struggle with cosets a little. How are you supposed to find the distinct cosets, especially whey you are working with a group such as D4? I get confused with the notation we did in class versus the notation in the book and what order you are supposed to do r^3d or dr^3.

2. I liked this section because it really did not seem like it was presenting any new information - it is based off the sections previous, the only thing different is that the groups are now normal. It is interesting to see the relationship between abelian and normal.

7.6 (part 2), due on March 24

1. As far as I could tell to check and see if a subgroup of a group is normal, you just take each element of the group and 'multiply' it on the right and left side of each element in the subgroup and see if you get the same thing. I think this is wrong though - if Na=aN, then this means na is an element of Na AND aN, not that na=an. I'll have to remember that...

2. I liked the proof of Theorem 7.33. I think the theorem will be useful in proving other things too, such as possibly with quotient rings in the next section. I think I am going to have to spend a little more time on Theorem 7.34 though, it was long but I am sure very powerful! :)

7.6 (part 1), due on March 23

1. I think I am getting confused between left cosets as presented in 6.1 denoted a+I, and left cosets as presented in this section denoted a+K. Is the only difference that I is a subring and K is a subgroup?

2. It appears that a normal subgroup N of G is similar to abelian (commutative) groups as talked about before, except now we are talking about the commutativity(?) of cosets. So Na=aN for all a in G.

Review, due on March 21

* Which topics and theorems do you think are the most important out of those we have studied? The three listed on the study guide that we will have to state and prove, as well as ones concerning rings, groups, and ideals.
* What kinds of questions do you expect to see on the exam? Some computational questions and some proof questions, similar to the last test I suppose.
* 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 on Monday. The second theorem we have to know and prove: If R is a commutative ring with identity and I is an ideal of R, then R/I is an integral domain if and only if I is a prime ideal.

7.5 (part 2), due on March 18

1. I think I am getting caught up on the meaning of cosets - we've talked about them before and I can do proofs with them in it, but the concept behind what they really are I am struggling with a bit. I think that is why the proof of Lagrange's Theorem and Corollary 7.27 does not to clear to me.

2. I thought it very clever how the table on page 204 was filled out - they did a case by case method until they got the right ones. I don't think I would have thought of this! The one on page 205 was a bit more crazy however...

7.5 (part 1), due on March 16

1. One thing that I didn't really like is that all of the proofs say, "Copy the proof of ... with the obvious notational changes" instead of giving the proof. I'd rather like to see the proof right there instead of having to look it up and adapt it.

2. I thought it was interesting that you can have cosets in groups - I guess I just never thought about it yet. That is good though because I am still trying to understand them better! It is also cool that you can have congruence in groups. It seems like you can have congruence in so many applications.

7.4, due on Pi Day!

1. I liked the first few pages of this section. But when it started into Cayley's Theorem I got a little lost - I don't really like having to dissect large proofs with weird notation :)

2. I have found that this textbook is loaded with material that is very complex. However, the more time I give myself to read each section, the more things I pick up on, and the more I learn. I think I just need to give myself more time for the sections. For example, I spent a good portion of 20 minutes on page 194 trying to understand the theorem and proof of Theorem 7.19, and I think I got it now. Also, the second example on page 93, I was confused with why they did not show that f(a)+f(b)=f(a+b) and then I realized that the operation shown in the book is not necessarily multiplication, it is the operation in G. Anyway, so time is good, I just don't know if I always can come by the necessary amount.

7.3, due on March 11

1. One question that I had was the example on page 184 - it says that the multiplicative group of units in the ring Z(15) is U(15)={1,2,4,7,8,11,13,14} by Theorem 2.11, but I don't see how this theorem would prove this. Then it says that 7 has order 4 in U(15) but they only checked 4 of them - or is that just part of the construction?

2. It seems like there are SO many more applications in groups than there where in rings! I hope I can remember them all, because there are a lot. One that I found interesting was the center of a group - it commutes with all other elements like how the identity element does.

7.2, due on March 9

1. I did not understand the proof of Theorem 7.8 on page 177 - maybe it is because I need to understand finite order better first. Hopefully we go over this more in class!

2. I am glad I took liear algebra before taking this class. There seems to be a lot of applications from it. For example, in Corrolary 7.6 there is an important distinguising feature between Abelian and non-Abelian groups: (ab)(-1) = b(-1)a(-1) and not always a(-1)b(-1). Like with matrix multiplication, (ab)(-1) = b(-1)a(-1) also.

7.1 (part 2), due March 7

1. I think I am getting lost in the notation - for example, Theorem 7.4 has a lot of notation going on that makes it a little more confusing to understand compared to something like Theorem 7.1 or 7.2. I think I am alright on everything else though.

2. When I read Theorem 7.2, I kind of thought of the set U of all units in R as a universal union - it seems like this concept comes up a lot in mathematics, especially in 290 and 341. It is interesting to see how properties from one mathematics transfers to other mathematics, which in turn unites them all!

6.1 part 1, due on February 22

1. I am not sure I understand what a "principle ideal generated by c" is. I am hoping we will go over this more in class...

2. The definition on page 135 reminded me of a commutative ring, closed under multiplication, only with a few stipulations: r has to be an element of our ring, a has to be an element of our subring I, and then ra, ar are both elements of I.

5.3, due on February 18

1. In theorem 5.10, I am not sure where they got a(x). Actually, I am not quite sure I even understand the proof. I was about to be fine with that since we will be talking about it tomorrow in class, but then I remembered I will not be there tomorrow, so I guess I'll just have to spend a little more time on the section to make sure I understand the material better. I do not want to get behind!

2. I liked the discussion on page 130 about the implications/history of theorem 5.11 - it reminded me of the things we learned about in History of Math (math 300). It is cool to see how Abstract Algebra deals with i's.

5.2, due on February 16

1. I think the most difficult part for me to grasp in this section was the notation F[x]/(p(x)). I keep wanting to think that / sign means divide, but really it means the set of all congruence classes modulo p(x). I guess it is something I'll have to get used to.

2. It is crazy to me that you can have modulo polynomials - I guess I don't quite understand how that is possible, but I still think it is cool. Not only can you have modulo integers, but polynomials as well!

5.1, due on February 14

1. I don't think that I like how all the proofs in the book just say "Adapt the proof of ____" from a previous section. I wish they would have shown more proofs so I could feel slightly more confident on the material.

2. It was way cool to me that the properties of regular modulo stuff that we have been working with has transferred over so well to polynomials so well! Even properties of equivalence classes has transferred over. So even though this is a whole new chapter, I am thinking I will be able to pick up on the tops somewhat easily.

9.4, due on February 11

1. I am not sure I understand why they define addition of equivalence classes as [a,b]+[c,d] = [ad+bc, bd], it does not quite make sense. I know you can randomly define addition as whatever you want, but how does this relate to similar fractions?

2. I liked going through this section because it applied everything that I am learning right now as I am studying for the test. It is interesting that they get such an intensive section all our of just a simple topic like similar fractions!

due on February 9

I think some of the most important theorems to know are the ones dealing with modulo, primes, the Euclidean Algorithm, and rings. As far as the exam, I am expecting to see all and every kind of question - I don't even know exactly what to expect as far as specifics go. Maybe some definitions, some statements of theorems, and some proofs. Before I take the exam, I need to get my theorems memorized and my definitions organized. I have learned a lot, but the definitions are kind of floating around in my head. So ya, I just need to study EVERYTHING!!!!

4.4, due on February 7

1. I did not understand what the distinction is and why the distinction was so important between the different representations of x in the statement f(x)=x^2-3x+2 and the the rule of the function f:R->R. How will this distinction help us? Also I did not understand corollary 4.18. I am thinking I just need to get a better grasp on what irreducable means...

2. I love how this stuff relates to easily to what we already know about polynomials - it is just way more intense than what we already have been taught - I guess this is why they call this class ABSTRACT Algebra. I am definitely okay with trying to find roots and factoring, so this stuff is pretty cool.

4.3, due on January 4

1. Something that I struggled with was trying to find all the irreducible polynomials of degree 3 in the integers mod2 WITHOUT writing out all possible options and cancelling out the ones that are reducible. I think there is a way that is simpler, like using associates, but I am not quite sure how that works.

2. What I thought was really cool was that you can apply properties of integers having a unique prime factorization, to what we talked about in this section. In other words, polynomials are reducible (like a prime factorization) if it has divisors other than just its associates (like the prime itself) and the nonzero constant polynomials (like 1). At least I am pretty sure that is the comparison. Knowing this was useful for doing the homework as well.

4.2, due on February 2

1. I am not sure how if f(x) divides g(x), then cf(x) also divides g(x) for each nonzero c in the field F, and how this statement shows that a nonzero polynomial may have infinitely many divisors... How can a polynomial have such a property?

2. The section was so cool - you could use the Division Algorithm on polynomials! This property as well as others like Thoerem 4.5 and 4.6 related directly to sections about divisibility in the beginning of the book. Thus, it was easier to grasp these slightly more difficult concepts with polynomials.

4.1, due on January 31

1. What I struggled with most was the proof of Theorem 4.4 - the existence of polynomials q(x) and r(x). It seems crazy that you can prove something like that with induction! Hopefully we go through it again in class so I can understand it better.

2. I thought this section was so cool! I love working with polynomials - factoring, multiplying, dividing, adding, etc. It seems like all the rules are familiar to what I have learned in the past, only now they are specifically denoted by certain things. Such examples include deg f(x), and R[x].

Response, due on January 28

I spend about 3 hours per homework assignment, so about 8-9 hours per week (not including the reading/blog post, which I spend about a 30-45 min on each). The readings helped prepare me for the lectures, and the lectures prepare me for the homework assignments. I am glad we are forced to read the textbook because otherwise I don’t think I would, and it has been very helpful. Working in groups and coming to your office to ask questions I still had about homework, even after having had lecture and reading the section has helped immensely as well. Probably the most helpful is the homework assignments – when I cannot understand a question, there are ways of getting help from you or others. This extra help has probably been the most helpful to my understanding.

Office hours for me are not possible for me to attend because I have a class. I sometimes come right before or right after class to ask a question or two, even though those technically are not office hours. I have really appreciated your time you have given to help me during these times – I feel when I can understand the homework, I can understand everything more thoroughly, and there is nothing more discouraging than not being able to understand the homework or material. So anyway, I hope it is alright when I stop by with questions. This is my only comment about the class thus far, other than I love it!

3.3, due on January 26

1. I was getting all confused with charts on page 67 and with the roman numeral discussion. I am not sure how we can prove matrices are isomorphic like on page 68 either.

2. What I did like about this section is that it talked about surjective and injective functions. I remember them pretty well from 290 and 341, so it is cool to see them again here in 371. I love it when the mathematics from different classes connect across the board! Life makes so much more sense that way.

3.2, due on January 24

1. I am still struggling with the concept of multiplicative inverses, what a field is, or what a unit is. All I know about mulitiplicative inverses is that if you multiply a number by its multiplicative inverse, you get 1. However, I am not quite sure how to find what that multiplicative inverse is.

2. I thought the first few pages of the section were very logical. It applied directly to basic arithmetic properties, so it was easy to understand. One thing that did not hold for basic arithmetic properties was that (a+b)^2 did not always equal a^2 + 2ab + b^2. I understand this because I understand 3.1 pretty well.

3.1 (second half), due on January 21

1. The second half of this section was rather short so I did not have much that I didn't understand. If I had to choose something though, it would have to be where it introduces subrings - part iv of Theorem 3.2 - where it says we have to show that the solution has to be an element of the set S, which I am not quite sure how to prove. Also, I do not know what a field is.

2. I thought it was interesting that they introduced something called "subrings". I know what sets are; I know what subsets are. I know what rings are; I am learning what a subring is. There are similar correlations between subsets and subrings, and it is interesting to see the comparison.

3.1 (first half), due on January 19

1. Although the section brings up over and over again the word "ring", I am still not quite sure what in the world that is. The textbook says it is a system that shares a minimal number of fundamental properties with the integers and the integers(mod n). I am guessing that is why they started talking about matrices and rings, because matrices are just another system where some of the fundamental properties with the integers is shared with those of the integers(mod n) like multiplication and addition are non-commutative rings with identity. I am still not quite sure what that means though, or what the book means by integral domain or a field. I am hoping that we will talk about it a lot in class - maybe that way I will understand better.

2. I did like it that they started bringing in matrices. I have taken Linear Algebra and liked the class, so naturally I like matrices. So ya, the connection between Linear Algebra and this class is pretty cool. I also think it is very interesting that they are bringing in the complex numbers.

2.3, due on January 14

1. This section was a bit more abstract for me compared with the other sections that I have read so far. I think it might be because there are just so many variables and I am loosing track of what each of their properties were. I am not positive I understand the logic of the proof of Theorem 2.8, it was just really long! I think once I do some of the homework problems this section might be a little easier to understand I hope.

2. Something I did understand and liked very much was the example on page 39. On the last homework assignment, it asked questions like this that we had to solve by guess and check a little bit, but this method is much more understandable and structured. I would like to try some problems like this on the homework assignment.

2.2, due on January 12

1. This section had many properties in it relating integer arithmetic to integer(mod n) arithmetic. I think the most difficult one for me to conceptualize was property 5 for modulo arithmetic: For each [a] in the integers(mod n), the equation [a]+X=[0] has a solution in the integers(mod n). I had to think a bit about what "X" would be, and I finally realized that it would be an equivalence class, more specifically it would be the equivalence class of [-a].

2. The most interesting part of the material was that you could take a large arithmetic problem regarding the integers in modulo n, and you can either compute the modulo 10 arithmetic and then simplify your answer to modulo n, OR you could simplify along the way so you don't end up with a huge number to reduce to modulo n.

Example (in modulo 3): ([4]+[7])*[2] = [22] = [19] = ... = [4] = [1] vs. ([4]+[7])*[2] = ([1]+[1])*[2] = [4] = [1]

2.1, due on January 10

1. The material was fairly simple to understand in this section, however the area that I may not thought about before is Corollary 2.5. It talks about the remainder having the same equivalence class as the dividend. I guess I am not quite sure why they brought in the t-s...

2. I loved the connection with the equals sign and the equivalence class equals sign. In both cases their properties had an identical outcome, and the properties of reflexive, symmetric, and transitive can be applied to modulo. I also find very useful that a=b(mod n) breaks down into n divides a-b which also breaks down to kn=a-b for some integer k. I like this because depending on the context of the proof, you can choose either of the three representations that will be most useful.

1.1-1.3, due on January 7

1. The most difficult part of the material was the material itself. I have seen these theorems numerous times in the past, but I have not been asked to rigorously prove them before. I am finding that it took me like 20min just to try to understand the Division Algorithm. However, I know understanding the material will just take time and practice. I don't think I fully understand any of the theorems... yet. But I will work on it! One thing that I did not understand about the division algorithm, for example, is why a-b(q+1)=r-b....

2. I am actually very excited to learn to prove such theorems rigorously! As I said above, I have seen these theorems (division and euclidean algorithms, and greatest common divisors) in many of my math courses, but I never quite understood the rigorous mathematics behind them. I have always wanted to learn how. Having a solid understanding of the proofs will greatly help me in my future math courses, and additionally as I end up teaching some day. The most interesting part of the material was in the section about greatest common divisors where it talks about the gcd being a linear combination of (a,b). I have never thought about it that way before.

Introduction, due on January 7

I, Sarah Hinton, am a Junior in the Math Education major.

The post-calculus math courses I have taken are Math 290, Differential Equations, and Linear Algebra.

I am taking this class to fulfill a requirement for my major in order to graduate. Additionally, this class will help expand my knowledge and understanding of mathematics.

One teacher that I had was not a very effective teacher in the sense that I probably could have learned more by spending that hour each day of class in the math lab doing my homework than by sitting through his lectures. During class he would write illegibly at lightning speed on the board and regurgitate the theorems and examples from the textbook. It felt like his goal was to get through the material, instead of getting the material through his students. I wish he could have been more personable and interested in finding out the level of understanding amongst his students (via asking if we understood, offering office hours, having a TA for the class, going a little slower).

I am working on memorizing the digits of pi, and I love to ballroom dance, cook, and play tennis. I do not have a cell phone or a facebook page. :)

MWF: 9-10, 10-11, or 12-1