#39 due Dec 7

Difficult- Section 37 seems to be pretty straight forward in theorems and examples. Sometimes I don't follow how the book comes up with certain things, but usually when I go back over it I can figure it out. 37.8 Lemma and its associated examples are the ones that cause the biggest challenge for me.


Reflective- I think 37.6 theorem is very interesting as well as 37.8 lemma.

#38 due Dec 5

Difficult- I have followed pretty much everything I think I am confused a little bit about how the conjugates work. I also think I need to review maximal subgroups since that is part of the definition of Sylow subgroups. But most of the rest seems to make sense it is just building on previous knowledge.

Reflective- Well I googled Cauchy to find out more about him and his theorem because I like mathematicians and he is cool. I also like this section a lot because it seems to have a large amount of repeated things like the normal subgroup and primes and order and everything.

#37 due Dec 2

Difficult- The Cartesian product of sets makes sense I am just really bad at remember exactly what the notation means. I am confused by 11.3 example because I don't see how that relates to anything new in the section because it seems like something we have already done. I do not understand 11.9 theorem very well. So I'm thinking I don't understand the section period. But then I see 11.10 example and that makes perfect sense. So I think it's notation and wording that is holding me up.


Reflective- Gotta love 11.5 Theorem because it goes back to isomorphisms and relatively prime. The historical note is interesting about how findings were finally turned into abstract theory.

#36 due Nov 30

Difficult- In the first isomorphism theorem I do not know what canonical means, but I don't know if that is important or not. In all of the theorems I don't follow what they mean exactly. But I do like the figures/diagrams because they are so much easier to understand!


Reflective- I think the 34.4 Lemma! The join term is cool and I think I understand it so gotta love it. The examples are so helpful in understanding what these theorems mean.

#35 due Nov 28

Difficult- There are two million and one symbols in this section! And so I am really confused what the exact definition of a factor group is and what 14.1 theorem means. The 14.2 and 14.3 examples help me understand what a factor group is a little bit more. 14.10 figure is so helpful! I was getting really lost with all the symbols and which meant what. I think the lecture will help me better understand these.

Reflective- I am glad we keep using homomorphisms and isomorphisms since we have used them and know a lot about them. 14.13 theorem we used last section so I know that one, yay!

#34 due Nov 21

So yeah I don't really know what I am supposed to be blogging about/if I'm supposed to blog? But I am going to do the exam questions.






Which topics and theorems do you think are important out of those we have studied?
Geesh there are lots. 7.8 Theorem is a big one because because it has 4 parts to it and they are very useful in a lot of proofs and problems we do. LaGrange's Theorem and its corollaries are also very important and are used a lot. I like 6.10 Theorem. 




What do you need to work on understanding better before the exam?
I am working on understanding what we did in class Friday. I also am working on proofs of Theorems because I always get lost in them. Just trying to go over everything again before the test!

#33 due Nov 18

Difficult- The evaluation homomorphism confuses me, but if I think of it loud and not use the symbols it makes sense to me. The phi symbol (at least I think that is what it is) with the numbers next to it like the trivial homomorphism, identity map, etc is confusing to me. I follow a lot of the examples but they are somewhat confusing.

Reflective- I like how the division algrorithm comes back into play again! I like 13.12 Theorem because it makes sense and is what you would assume. Also the historical note is interesting as it is neat to read about people who have discovered things, such as cosets being the same.

#32 due Nov 16

Difficult- Omgsh I do not like this notation! I am trying to make sense of 11.2 Theorem and I think I get it but it is a little bit confusing. Understanding the definitions and theorems are taking me a while to follow because I keep having to stop and make sense of the notation. When I see the examples I understand more what the definitions and theorems are saying.

Reflective- I think that 11.5 Theorem is very interesting! It is neat how Zm x Zn is cyclic and isomorphic to Zmn if m and n are relatively prime. As always the historical note is interesting. In general I just like the abelian concept so I like that part of the section.

#32 due Nov 14

Difficult- Well notation is always confusing to me but I think I understand the relation statements in 10.1 Theorem. However in 10.7 Example I do not follow what is going on! I think I don't follow the index definition and why it is the number of just the left cosets. Maybe because the right has the same number so they just picked one? I don't know if that's even important.

Reflective- Well we have talked about left and right cosets previously so I remember those and they make sense. The Theorem of LaGrange is logical and makes sense!

#31 due Nov 11

Difficult- Okay I think I get these and I like them but I am lost in 9.10 example and how they are coming up with the permutations so I guess I need to figure out how they are multiplying cycles if they are not disjoint. I also do not understand transpositions, which I feel like should  be easy. But I don't get 9.13 and 9.14 examples.

Reflective- It is interesting that permutation multiplication is not commutative but but multiplication of disjoint cycles is. It is interesting that 9.15 Theorem has two very different proofs and different methods.

#30 due Nov 9

Difficult- The permutations seem pretty straight forward and easy to understand. I do not know the word lemma so that is new to me. Towards the end of the section, with Cayley's theorem all of the symbols and notation start to really confuse me. However I think I understand a little bit so hopefully this section will not be too bad!

Reflective- Of course I always like the historical notes! And this one is indeed very interesting about permutations in Islamic and Hebrew cultures and their alphabets. Also it is very cool that S3 is also the group of symmetries of an equilateral triangle.

#29 due Nov 7

Difficult- Well as always the notation seems to be confusing!! Also I don't get 7.6 theorem in words I don't think but the proof seems to make sense and I can follow that so maybe I just need some explanation. At first I was confused by the example 7.7 of the Cayley graphs because they were two different shapes but then it says that doesn't matter so never mind about that but in figure 7.9 I guess I don't get the relationship between all of the vertices and how they are connected.


Reflective- The subgroup generated by ai of the intersection is like the same of the cyclic group generator so that is nice and consistent. The Cayley graphs are very interesting! I think I like them, well the ones that I get that is!

#28 due November 4

Difficult- I was following along fine until I got to 6.10 Theorem. I think I understand it somewhat, they are proving it is an isomorphism and I get that. I don't understand what is meant by "describing cyclic groups UP TO an isomorphism" but then the prove the isomorphism. Example 6.17 seems confusing but the book says it is straightforward and I think I am making sense out of it now.


Reflective- It is interesting that the division algorithm, the greatest common divisors and relatively prime come back into play with cyclic groups; a lot of things repeat themselves in this class, this is good. I like how the section ends on page 65, because I was just thinking that that example looked complicated.

#27 due October 30th

Difficult- I think I am struggling with the identity in groups. On the top of page 52 there is a subgroup diagram but it does not make sense to me. I don't know if there is a difference between a^(-1) and e or if they are the same inverse for a group or subgroup.


Reflective- The cyclic subgroups can relate back to principal ideals because they were generated by one element, as these cyclic subgroups are which is neat.

#26 due October 28th

Difficult- I think I am terrible at understanding this notation! It is kind of confusing. I don't think I understand the binary operation and what that means. I am definitely going to need to get this during the lecture! Also the group tables seem to be a little confusing. I think everything in this class confuses me.


Reflect- The historical note on page 38 is interesting because it relates the groups to other types of math like geometry and some of calc 2 or 3 I believe with LaGrange and permutations. I also like the historical note on page 39. They just add something cool so the text isn't as boring!

#24 due October 24th

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

We have done so much this section geesh! Well I feel like the division algorithm theorem is very important which is number 4.1.7, we used that one a lot. Then the factor theorem is also important. The extension field, K, theorem is important and also cool. And then we learn what an ideal is and we have used it a ton since then so that is important. In what we have done lately the isomorphism theorems have also been very important.

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

I think I need to work on understanding notation, and how to understand what questions are asking me. I will need to review homework problems and examples to know how to better do that. I also need to work on how ideas and theorems relate to each other so when constructing proofs and such I can actually do them. Proofs are what kill me.

#23 due October 21st

I think I am confused on the blog schedule. It says for today to read section 26 but I think that's what we did in class yesterday! So I'm blogging on section 27.

Difficult- Okay I think I do not remember what a proper ideal is so I do not understand the definition of a maximal ideal of a ring. I will look that up though so I can understand this. I think I follow everything else. I am getting really worried for the test though. I also need to figure out what proper nontrivial ideals are.

Reflective- Abstract algebra is really cool the way that primes play into it. Prime ideals seem to be neat. If I can understand them correctly!

#22 due October 19t

Difficult- As you mention, the notation with all of the plus signs is confusing. It requires slowing down and thinking about it for it to make sense, which is fine it just isn't easy right away. I think I am still confused also on what the notation R/I means. Once the lecture turns into the isomorphism theorems with the quotient ring I get very confused. I don't know if maybe once I see the theorems worked out if I'll get it or not but it looks to be really hard.

Reflective- It is neat that the quotient rings were not intended to have anything to do with homomorphisms and isomorphisms yet they do. And it is also cool that we're bringing homomorphisms and isomorphisms back into lectures. The old stuff makes me feel like I at least know something  because this new stuff is confusing to me! I will definitely be in before the exam with questions :)

#20.5

(end of 20)

Difficult- I don't understand how you would do examples 6.1.13-6.1.15. I think I understand the paragraph about principle idea but then I do not get how you do problems with that. The rest of the lecture about congruence seems to make sense though. Cosets might confuse me a little but I think I can figure them out.


Reflective- I think this ideal stuff is cool because it ties together more stuff that we have done previously like subrings, congruence and mod stuff.

#20 due October 12th

Difficult- Okay the definition of the word ideal is confusing. Although I remember you starting to talk about it in class on Monday so it seems familiar and the examples make more sense. I think of all the symbols confuse me . Reflective- The idea of ideal is pretty cool and there are a lot of examples so I think it will be a big topic. This section doesn't seem too bad and I will probably understand it more after class.

#19 due October 10th

Difficult- I don't think I understand how to form the extension field of F or where that comes from. Another part that confuses me is how x^2 + 1 has in Z7 has 49 elements. I don't see how it could. But the notion that a polynomial in Zp with degree k in a filed F[x] has p^k elements is interesting.

Reflective- I think the most interesting part of this lecture is example 5.3.2 which shows [x] is a root of that polynomial in the extension field K even though it is irreducible in Z2. I literally said cool out loud when I read this. This lecture is really interesting and I like it.

#18 due October 7th

Difficult- Well this lesson seems to be pretty straight forward like the ones lately have been. The only thing that seems confusing is that F[x]/(p(x)) contains a subring F* that is isomorphic to F along with the paragraph describing it on the bottom of page one. I know that it gets reworded later but it still kind of seems confusing to me.


Reflective- I am still glad we are using stuff we've seen before in a different way. Really it makes me less worried about this class. Of course I still worry, just not quite as much. I always worry about getting the homework done!

#17 due October 5th

Difficult- Starting at about corollary 5.1.11 I become confused and I don't understand what that corollary is saying or the examples that go along with it. Also definition 5.1.6 makes sense but then I don't get the examples and how you would figure them out. And lastly I don't think I know or remember what disjoint and identical mean.


Reflective- Well once again more stuff we have done is coming back and I like it since I understand it! Well some of it that is but at least it looks familiar. And I especially like that the proofs are basically the same since those are always a challenge for me.

#16 due October 3rd

Difficult- Corollary 23.6 was the only confusing part of this section. I am not quite sure what cyclic means or what the group <F*, .> means and I don't think I get how that relates to the factoring and what we've been doing.

Reflective- I like the factoring part because that is very basic and easy to catch on to. I also like long division so I'm glad we keep using it.

#15 due September 30th

Difficult- I followed along pretty well until Theorem 23.15, the Einstein Criterion. I don't get the example or corollary that go along with it either. And then the whole uniqueness of factorization section I don't get as well. A lot of the theorems in this section seem to run together for me, I don't think I'm understanding the main idea of the section.


Reflective- I think the irreducible concept is pretty cool. I followed a lot of the first part of the section and think I can do some of the problems. Something interesting about this section is they leave out or summarize proofs which I thinks shows this section is hard!

#14 due September 28th

Difficult- Well this all relatively makes sense I think since I have seen it before. The trick will be figuring it out using polynomials but that doesn't seem to be too bad. I think I am confused about what a monic polynomial means and when it is monic and when it is not since that's mentioned with the Euclidean algorithm.

Reflective- I think it is great all of this stuff transfers over to polynomials in a field since we have done it before and it makes sense! I also liked this section a lot from last time so I'm glad to see the same things just now with polynomials. I am excited to do the homework... weird.

#13 due September 26th

Difficult- I am confused! I do not understand all the ways a polynomial is different as a ring in Z[x]. This section is like a foreign language to me. There are so many symbols with the sums and then with the mapping. I do follow however what the division algorithm is saying and dividing polynomials so at least I get something. I think understanding the polynomials will take me a little bit but hopefully I can get it.


Reflective- I think the historical note on page 198 is interesting. I think it's cool how the book includes these. It's weird that around 400 years ago x and letters toward the end of alphabet were used to represent indeterminates and this is still true today. I also like the division of polynomials in this section since we did that a long time ago in and it's fun.

#12 due September 23rd

Difficult- I may have spoke too soon on the last blog. I don't get today's section. The start of the section should not say it is quite clear how a structure- relating map of a ring R into a ring R' should be defined because I don't see how it is clear! Okay first that the phi symbol is a map. I remember maps from previous classes, but I don't get what def.18.9 is saying. I am also confused about the abelian groups and how to interpret what <R, +> or <R',+> mean. I remember the terms kernel and isomorphism from previous classes also but I don't follow how rings are relating to these. I also think the examples are confusing. I really hope I get this in class.

Reflective- I don't like this because I don't get it! This section is practically all symbols which is not enjoyable. However looking at example 18.13 helps me to better understand homomorphisms because the example is in numbers and that example makes sense to me.


Questions
1. To study for the exam I went through each lecture we covered. I read the lecture, my notes on that lecture, and looked at the homework problems, especially the ones I missed. I then made note cards about the items on the exam checklist that were covered in that section, and about other things I thought I should memorize. I did this for each section then I started memorizing my note cards. This really helped me to know exactly what things were.
3. Maybe more examples. Some things don't make sense to me until I see different examples of them then things fall into place more.

#11 due September 21st

Difficult- I think I am understanding more! I followed the the first couple parts of the section... and they made sense! I am very excited about this. However, towards the middle of the section (incorporating the fields) I got confused. Theorem 19.11 is the one I find the hardest to follow. I think it is one of the ones I need to hear explained out loud and hopefully I'll get that. The rest seems not too bad. But I haven't done any exercises yet so hopefully I'm not speaking too soon!


Reflective- I thought the Venn diagram was very useful. It helped me put a picture in my mind of how the different types of rings relate to one another. I like looking at integral domains and different rings that can and cannot be integral domains. The idea and the fact that I get it probably contribute to why I like them. For the same reasoning, I like the divisors of 0 and cancellation laws.

#10, questions. Due September 16th

One of the most important theorems is theorem 1.2.7 because it is very useful when solving proofs in multiple ways. Theorem 2.1.5 is also an important one that helps in several proofs because it connects modular world to integers. Although we have only seen this theorem for a couple of days I believe theorem 2.3.5 is important because its 2 conditions will help when solving proofs.

I expect to see a lot of questions that take the definitions one step further so that we show we actually understand the definitions. We will have to solve problems involving these definitions to show understanding and how to use the definition. I also expect proofs that tie together multiple theorems so we demonstrate understanding of those and how we can use them together to solve problems.

#9: Lecture 9 (and 10?), due September 14th

Difficult- Well I was reading through the first page of ring stuff (in the lecture) and I think I understand the axioms and the different types of rings. But then when I got to the examples I had no idea if it they were rings or not. I found the book to be equally challenging. I am starting to get really nervous for this test now with this stuff included also. I think I will need to see examples in class and I will hopefully be able to figure out rings and then the fields.

Reflective- Well the ring stuff seems pretty cool, what I get of it at least. When I figure it out for sure I think I will like it even more. I liked the history section in the book.I think it's great that a woman mathematician helped to develop rings and was able to become a professor! So if she can do it, then hopefully I can too!

#8: Lecture 8, due September 12th

Difficult- I think I am getting confused between the equations and the solutions. Okay so I think that if I'm following this correctly, in Zn if n is prime there is a unique solution to the equation ax=b, if a doesn't equal 0. But if n is not prime then only some equations have solutions? In that equation am I supposed to assume that a and b are classes since there are no bars? Because then in the next corollary the equation is back but with bars. Do they mean the same thing? Okay actually I think the second one might be different because it says a and n must be relatively prime.

Reflective- I found it interesting that in Zn if n is prime then there are special properties. I really think the Zn world is cool! When we were doing the tables in class I liked those a lot. Now it is interesting to see when n changes how properties and equations are formed, such as when n is prime. I also liked bringing the Euclidean algorithm in because I understand that so I can follow what that part is saying.

#7: Lecture 7, due September 9th

Difficult: Well I am hoping this lesson is not too difficult for me because I somewhat remember the scalar addition and multiplication and the properties from Linear Algebra. The tables for Z5 and Z6 I find confusing however. I should be able to figure them out I think because I do recall some of this stuff, but I'm not following where the 0s come into play for addition. I see the pattern in the table but that's obviously not the only way to come up with the numbers. I might get confused when proving theorem 2.2.3 (the properties) but hopefully I can get it down.

Reflective: I don't feel as confused after reading this lecture as I did the last one! I like the modular arithmetic in the beginning of the lecture because that makes sense to me. I can clearly see how a number in one set added to a number in another set equals a number in another set.And it makes sense why the addition would need to be well-defined. I like seeing things this way. Because when it comes to proofs I get lost.

Answers to additional questions:
1. I spend probably about 30 minutes- 1 hour on the homework. The lecture prepared me for them mostly yes. The homework is getting harder though so I'm starting to struggle though and take longer. You should expect a visit from me in the next week.
2. I hate proofs. And that's what this class is. I am super worried. I follow them when you do them, but most I don't know how to formulate and think of the way to do them on my own. I also dislike that everyone has had analysis so I feel like they're at an advantage. Examples contribute most to my learning.
3. I think the class is great, minus the material. That's something you can't change though so I just have to try my best to understand it all.

#6':Lecture 6', due September 7th

(Difficult) I am not quite sure what a modulo is. How does a modulo relate with a congruence class? Does the equals sign but with another straight line over it mean congruency? I was confused after reading the first set of notes but when I got to the second set it cleared some things up. Now I don't know if I get Theorem 1.3.5. And I think I'm missing something big here... how are primes related to congruence classes/modulos? (Reflective)  I don't think I understand a lot about primes. Because I was unaware that every integer except 0, -1 and 1 is a product of prime numbers and how to do the prime factorization (Theorem 1.3.4&5). I also did not know that p is prime if and only of -p is prime. However I did know that there are infinitely many primes so there's something! But how to prove that... hmmm I don't know. I guess I will learn more about primes in class!

#6:Lecture 6, due September 7th

Difficult- Well I think the most difficult part for me was keeping track of the steps during the euclidean algorithm with large numbers to find u and v. I think during the lecture I was a couple of seconds behind because it was taking me a little a little longer to follow each step. The other hard part for me was the proof of theorem 2.7. There were also several steps in this proof and once again it would take me a couple of seconds to figure out what exactly we were doing in the proof. I worry that if I had to prove something this big on a test I will struggle because there was so much to the proof.

Reflective- I really like the division algorithm. And even though the large euclidean algorithm took me a bit, I really like it doing it also. I like the computation and I like how it works out every time. It also is pretty easy so that's a reason to like it even more. I actually want to do this homework also, so this is a very good thing.

#5: Lecture 5, due September 2nd

Difficult- I am following what the lecture says in words, but the symbols are confusing me. I don't know what the c with an line underneath it means. I understand what each of the properties mean with the equivalence relations, but when applying them to sets I am lost. Sets were not my forte in linear algebra so I am nervous. I hope when we discuss this in class it makes more sense to me. It also helps to think of sets in words instead of symbols so some the examples make a little more sense.

Reflective- Well I like the reflective, symmetric, and transitive properties since I learned those forever ago. I like that I learned them a long time ago but they play a big part in many areas in math so when they come up I know what they mean. I also think the cross products of sets interesting because that is how you form new sets, taking elements from both sets to make a new one.

#4:Lecture 4, due on August 31st

Difficult- Okay well even though we covered this lecture in this class today, I think I am still confused on which technique to use when. I am going to try the homework for it and see if that helps because I know you said practice is what helps you master the proof techniques. I understand the techniques of proofs 1 but 2 is a bit more challenging for me. I am still worried I will not know what technique to use when I'm given an implication. The most difficult technique seems to be proof by induction. Theorem 1.7 and Example 1.8 were hard and took me a while to follow. Hopefully once I practice these I can get the hang of it.

Reflective- The most interesting part of the lecture to me is needing to prove 2 cases. I think I might like this one because it seems to be the most obvious proof to solve. The implications that require 2 cases are pretty straight forward, or so I think, so I will hopefully be able to figure these out. I also like the well-ordering axiom because I  clearly understand this one. Most of the rest of the material I have to think about for a little longer but the 2 cases and well-ordering axiom are interesting and I understand them.

#3:Lecture 3, due on August 29th

Difficult- In symbols and descriptions I follow exactly what the lecture is saying. Our applied linear algebra class started out learning a little bit about contrapositives so that came back to me. However when it comes time to prove a contrapositive and the examples are in sentences, I struggle. This is weird to me because with lectures one and two it was easier to think in sentences than in symbols. I can do the contrapositive example but it took me a while.
I am wondering on a test how will I know when to use the contrapositive, are there words that signal to use it? The contrapositive example given is an "if => then" problem and an example of deductive reasoning is an "if => then" so how will I know which is way is the correct way to prove a problem? Trial and error?

Reflective- The most interesting part of the material to me is that the inverse of the implication is equal to the converse of the implication, therefore the inverse is the contrapositive of the converse. I had not learned that before when we had briefly discussed this is in linear algebra but it makes sense and connected the different parts of the lecture. Making the truth table helps to solidify for me that the implication and the contrapositive are equivalent, and that the inverse and the converse are equivalent. I think I will get this even more when I do the homework questions that go along with this.

August 25th, 2010. #1 Introduction

1. Well, I am in my third year of college but am technically a senior by hours, and planning to graduate a semester early. My majors are Mathematics, Secondary Education, and Applied Critical Thought and Inquiry.
2. Linear Algebra is the only post calculus class I have taken, and this semester I am taking Abstract Algebra and Differential Equations.
3. Well besides the fact that I have to take this course, I am excited to learn about ring, groups, and fields. My pastor majored in Math and so he has told me a little bit about it. He also said that it was the most difficult math class he had to take so I am very nervous! I am not very good at proofs :/
4. Well Dr. Nicholson is the best math teacher I have had. He always had his office open so I would go in at least twice a week to get outside help. It helped me a lot to talk one-on-one and work out what I wasn't understanding. Least effective for me is not being open to help students outside of class.
5. Well something interesting. Hm, I am from Kansas and my family and I are HUGE Jayhawk fans. Rock Chalk!
6. Unfortunately, I cannot come during your office hours. I have from 10:15-1:30 open MWF so I hope sometime in there works for me to come get help because I am telling you now I will need it!! I hope this is okay. 

I am excited for this semester and getting to know you!