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.