Section 4.4, due on September 29

The most difficult part of this section was the the idea of a polynomial inducing a function. I have never heard of this before and it is still not super concrete in my mind. I'm not really sure why the concept of a polynomial inducing a function is so important and why it needs to be so distinctly distinguished between polynomial. I also found Corollary 4.19 to be a little hard for me to understand.

The most interesting part of this section was the remainder theorem. I don't think I had ever realized that before and I thought it was really neat. I also thought it was cool that to show that a deg 2 or 3 polynomial is irreducible you need only verify that it has no roots in the field F. It reminded me a lot of high school algebra, but with a cooler twist than they teach you. I feel like I finally am starting to understand the origin behind some of those facts that I just took for granted.

Section 4.2-4.3, due on September 27

The most difficult part of these sections for me was remembering all of the implications of Theorem 4.2. It doesn't seem like that hard of a thing, but I had trouble remembering when a degree of a particular polynomial is less than or equal to another. In section 4.3 the hardest thing was figuring out and applying the definition of irreducible and associat element. This were use often in proofs and these concepts are still a little fuzzy for me. I had an especially hard time following the proof of Theorem 4.11.

The part of this material that I found to be most interesting was the fact that a nonzero polynomial may have infinitely many divisors where, in contrast, a nonzero integer has only a finite number of divisors. I also thought is was interesting that in an integral domain, a unit is a constant polynomial yet in a field a unit must be a nonzero constant polynomial. The conditions that apply to fields are a little more strict, which I thought was interesting.

Section 4.1, due on September 24

The most difficult part of this section for me was following the proof of the Division Algorithm in F[x]. The subscripts and different notation was a little hard for me to follow. The whole concept of P containing an element of x that is not in R, yet with properties related to R is a little confusing. I guess I don't really understand why this idea is necessary in order to define a polynomial.

The most interesting part of this material for me was now much it reminded me of my Differential Equations class with leading coefficients and degrees and such. I also thought it was interesting when they started talking about x being a specific element of the ring P, and not a variable that can be assigned values. It was also interesting to note how closely the Division Algorithm in F[x] relates to the previous Division Algorithm that we learned.

Exam 1 review, due on September 22

I think that some of the most important topics to study would be those concerning rings and modular arithmetic. Having clear how to show something is a ring, subring, field, integral domain, isomorphism, homomorphism, etc. seem like topics that will be important to know. So basically all of the theorems and definitions that relate to these topics are areas that I think will be important. I also think knowing modular arithmetic properties will be especially important because it applies to divisibility, primes, and rings. Modular arithmetic seems like something that connects together everything that we have learned.

On the exam I expect to see lots of definitions and about 2 theorems out of the list provided that we will have to prove. I also expect 1 or 2 other proofs that will be very similar to previous homework problems that we have had. I expect something about how to show that a particular set is a subring, isomorphism, and homomorphism whether it be a proof or just giving the definition.

Section 3.3, due on September 20

The most difficult part of this material for me was following the isomorphisms from a set a integers modulo n, to a Cartestian product. I also found it hard to comprehend exactly what was being described by isomorphism and homomorphism. I think that I understand, but this was a new idea for me and sort of difficult to follow. I am also finding that many of the past couple of theorems are very similar to past theorems as they both deal with certain properties that allows something to be defined as it is. It has been a little difficult to try and keep all of these straight in my mind.

The most interesting part of this material was the fact that certain properties are preserved after isomorphisms. For some reason I just thought that was a really cool concept. It serves as a good analogy for many other aspects in life. I also thought that structure being preserved from one ring to another was a cool concept in general. It reminded me of chemistry classes I have taken where sometimes certain reactions will preserve properties of elements, while other times things aren't preserved at all.

Section 3.2, due on September 17

The most difficult part of this section for me is remembering what can and can't be assumed for different rings. I also had difficulty following the proof for theorem 3.11. In this section in particular there is so much manipulating of equations, that at first it is hard to see where they are going or why they are doing what they do. This makes it a little confusing and hard to follow at times. The term "unit" is also new and confused me a little because I would always call a unit just a number with a multiplicative inverse.

The most interesting part of this material for me was Theorem 3.6. I found it interesting that just by having multiplication and subtraction that the process of showing that S is a subring of R is reduced down to 2 steps. I also found the relationships between integral domains and fields to be hard to follow, but really interesting at the same time.

Response to questions, due on September 15

The homework assignments have usually taken me between 2-3 hours. The lecture and reading definitely prepared me for them, it's just that the homework requires putting concepts together in ways that I usually haven't thought of quite yet.

I have really liked the fact that we are required to read before class. This has really helped me to understand the material better. I also like that we can still ask questions and turn the homework in by 4:30 pm of the day that it is due. This has helped me because I feel like I can get most of my questions answered before I finish the homework.

To help me learn more effectively I just like to see examples of the concepts being applied and how they connect to one another. Sometimes an overview sheet of all that we have learned is helpful, or even mini reviews from time to time. This class has built on past concepts for the most part though, which has definitely helped it all stick more than otherwise.

Section 3.1, due on September 13

The hardest part of this section for me was keeping track of all of the different axioms and definitions as they would explain the properties of various rings. It was harder to understand the more abstract types of rings because I was not as used to the symbols being used in the ways that they were. I am also not really quite sure I understand exactly what a ring is and why it is something useful to have defined.

The most interesting part of this material for me was to see how much it reminded me of vector spaces and subspaces from Linear Algebra. It was also interesting for me to see which axioms held for different types of rings, and which properties had to be proved in order to show that something is a subring. It makes me wonder what sort of applications various rings have in the real world and if they are widely used and put into practice during current times.

Section 2.3, due on September 10

The most difficult part of this section for me was following the steps of the proof to Theorem 2.8 as it showed step 1 implied 2 and step 3 implied 1. Following the switch of notation during the proof was a little difficult for me because it made me have to remember exactly what they were talking about as I followed the proof, whether it be numbers or congruence classes.

I found the second part of Theorem 2.8 to be especially interesting. I just had never realized that and so it was really cool to learn. I also thought that Theorem 2.11 was a handy pattern that has been discovered. Both Theorems 2.8 and 2.11 reminded me of definitions/theorems in Linear Algebra regarding invertible matrices which was an interesting connection. It made me wonder if there is any sort of use to putting the multiplication/addition tables of the different congruence classes modulo n, into a matrix.

Section 2.2, due on September 8

One of the most difficult parts of the material for me was trying to understand their definition of addition and multiplication in the set of integers modulo n. Theorem 2.6 makes sense to me until I start thinking more about the discussion preceding it. If 1 is in B and 7 is in C, then B+C=8, so the equivalence class containing 8. Yet if -3 is in B and 15 is in C, then B+C=12, so the equivalence class containing 12. So I don't understand why 8 and 12 aren't in the same congruence class? I also find it difficult to remember which modulo we are in and adjust my arithmetic accordingly.

The most interesting part of the material for me was Theorem 2.7. These properties that are valid for the set of integers modulo n are very similar (probably the exact same) as the axioms that must be proved in Linear Algebra to show that something is a vector space. It is interesting to me to try and think how the set of integers modulo n and the concept of congruence classes will be used in future sections.

Section 2.1,due on September 3

The most difficult parts of this material for me were understanding the definition of the congruence class of a modulo n. Congruence and modulus are fairly newer concepts to me (as far as using them in proofs and other mathematical problems) and so keeping track of what they actually mean seems to be a little tricky for me. Because of this, I also found the first part of Corollary 2.5 to be a little bit harder to follow because I had to do a couple of examples and convince myself that it actually is true as stated.

The most interesting part of this material to me was Theorem 2.3. Once I came to understand the congruence classes, I thought that this theorem and its proof were pretty neat. The patterns found in all of number theory related topics is really rather clever. The modulus is something that came in handy when I took a computer programming class. It was really useful in writing code when you wanted items to appear on the screen in a patterned way (like 4 items for every line). I have also heard, but am not sure, that modulus is used a lot in RNS cryptography and I find that to be really fascinating.

Section 1.1-1.3, due on September 1

Some of the more difficult parts of the material for me included the proof of the division algorithm. Towards the end of the proof when we are showing that q and r are unique (bottom of pg. 5) I am still a little confused as to why we can just add those two inequalities to further the proof.

The other difficult parts of the material for me were convincing myself that the theorems and lemmas actually worked, and understanding what they were actually saying at the same time. For example, Lemma 1.7 took me awhile to process what was actually going on. I also found it difficult to follow the Euclidean Algorithm and recognize the pattern, keep track of all the subscripts, and retain what it was saying. In general, as equations are written in different forms and we say x divides y and so on, it is difficult for me to keep track of what is the dividend and what is the divisor.

While this material was the most difficult for me, I also found parts of it to be the most interesting. I really think Lemma 1.7 is cool as it basically states that the g.c.d. of a dividend and a divisor is the same as the g.c.d. of the divisor and the remainder. This was a really interesting pattern to me. I also found it interesting how Theorem 1.5 and 1.8 related in that they were both provide specific cases which answer the question: If a divides bc under what conditions is it true that a divides b or a divides c?