#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.