6.3, due on Wednesday, March 2

1. The most difficult part of this reading was understanding how the term prime can be applied to ideals. I do not understand how an ideal is prime when if b*c is in it then either b is in it or c is in it. Why does that make it prime? Then there is a theorem that says that something is a maximal ideal iff the quotient ring is a field, and yet on our homework it asks for the maximal ideal of rings that are not fields, so how does that work? I am so confused!

2. This reading connects to ideals to primes, which is really cool (once I am able to understand it!). I have only seen primes as associated with one number, but now we are calling an entire ideal prime! It is weird and confusing but still cool at the same time. I wonder if we will continue as the sections go on to add prime properties to things such as ideals and other topics in this abstract algebra course.

6.2 Part 2, due on Monday, February 26

1. The most difficult part of this section of the reading is keeping all the theorems and ideas straight with the kernels and quotient rings and regular rings and ideals and etc. There is so much to remember to then apply it to problems that are on the homework. I am so confused about which one of those terms when paired with another makes a function isomorphic or homomorphic (like theorems 6.12 and 6.13). It is all so unclear and the proofs do not help me understand them so I just keep getting more confused.

2. The most interesting part of this section is how kernels are being used in theorems to connect our rings back to homomorphic and isomorphic. I would have never guessed that the kernels we saw in linear algebra would come back and be applied to something we did at the beginning of the semester in Abstract Algebra. I love how connected mathematics is!

6.2 Part 1, due on Friday, February 25

1. I am still struggling with finding the cosets and how many there are and what they are etc., so I think I am going to struggle with this section because it deals more with cosets and the multiplication of and addition of them, but I can't even find them to use them yet. So there is no way that I can do addition and multiplication tables of cosets when I can't even figure out what the cosets are. So this is the most difficult part of this section -- the fact that cosets still make no sense to me.

2. It is interesting that R/I is a ring. I had no idea from the last section that this would be coming. I can see how a coset is like a congruence class, but to have R/I - the set of cosets - be a ring, well, that is just really cool. I like seeing how things that make no sense in one section (like R/I) are connecting to a big principle (like rings) that we have been working on all semester. I like when weird things turn out to have a purpose. It is cool.

6.1 Part 2, due on Wednesday, February 23

1. I do not understand what a coset is. It does not make sense to me. I do not see why we need to write a congruence class as a + I for it to be the formal symbol. It is just confusing me.

2. It is interesting that all the properties with mods and being an equivalence relation are applied to an ideal. I figured that would have been trivial and obvious. So it is interesting that the book goes through specific theorems and proofs to show how things work in mod I.

6.1, due on Tuesday, February 22

1. The most difficult part of this reading was that I have never even heard of an ideal before. Most of the time the new terms in this book I have at least heard of so I know that they are important. I have never heard of an ideal so I have no idea where learning about the concept of ideals will take us in this course, so this makes it conceptually very difficult for me to learn about them when I do not know what they purpose it.

2. The most interesting thing in this section was how in order to prove if something is an ideal, you use similar properties as you would in proving something is a subring. That is interesting because when I first read the definition of an ideal, I was not thinking that is was similar to a subring, so I like that there are connections with these new ideas and theorems and that some things in math are similar.

5.3, due on Friday, February 18

1. The hardest part of this section is understanding how to use the principles and theorems and apply them to actually proving if something is a field. I was starting the homework for this section and a lot of the problems ask you to prove is something is a field. I did not know how to apply what I had just read with the theorems to be able to compute a proof to show is something is a field. So this is the most difficult part of the reading.

2. I think it is really cool that Theorem 5.11 helps to explain problems and doubts that people had back in history with complex numbers and negative numbers. It is hard for me to connect Abstract Algebra with normal, useful math. I don't feel like it has a place in mathematics. However, this section showed me how Abstract Algebra is used to explain simple math concepts like negative and complex numbers -- things that would not be as common had there not been Abstract Algebra and this theorem to help people grasp the concepts.

5.1, due on Monday, February 14

1. The most difficult part of this reading is not the new theorems and properties, because they are things we have already seen before with mods just applied to functions and polynomials. The most difficult part of this section came from looking at the homework problems and realizing that I have to remember all the other theorems and properties and concepts we have discussed up until now and apply it to this new twist on the topic (like having to remember how to list different congruence classes and apply that to polynomials and the properties of the nonzero constant polynomial). That is hard because I do not remember things or memorize quickly. I have to look in the book and there are too many sections to do this to remember things to use for the homework. So this is the most difficult part.

2. The most interesting part of this reading is seeing mods and congruence classes applied to polynomials and fields. I would have never seen this coming, but I guess this has happened a lot in this course - the building on of the same theorems applied to different things - that I should have expected polynomials to come back into congruence classes.

9.4, due on Friday, February 11

1. The most difficult part of the reading was keeping straight all the new notions, with things like [a,b], (a,b), ~, and a/b. I am really bad with new notation and remembering what it all means and why, so that is the most difficult part of this new section.

2. It is interesting that this new material uses the terms of an integral domain and such, which I have never seen before this class, and connect them with equivalence relations, and the "~" symbol and meaning because we talked about that stuff in Math 190. But it has been awhile since I've had that class, so I don't remember a lot, but I do remember we did stuff with it, so it is neat that it connects to Abstract Algebra and integral domains.

Review Questions, due on Wednesday, February 9

1. Which topics and theorems do you think are the most important out of those we have studied?
I think the topics of rings, homomorphisms, isomorphisms, and fields are most important. So knowing the definitions and important theorems and properties about each of these topics is probably really important. It is also probably good to know about the Euclidean algorithm.

2. What kinds of questions do you expect to see on the exam?
I expect to see questions that require you to know the definitions and use them to prove theorems similar to ones we did in class and on the homework. I expect the questions that are more algebraic in nature to be like the ones we did on the homework.

3. What do you need to work on understanding better before the exam?
I just need to review all the definitions. I understood the material when I was doing the homework, but it is recalling it on the spot and without notes that is difficult. So just reviewing all the elements that make something a ring, integral domain, field, etc. is what I need to study for the exam.

4.4, due on February 7

1. The most difficult part of the reading is how to determine how to find a root for really hard polynomials and/or mods. The roots are the biggest part of this section and the theorems make sense, but applying how to figure out if something is a root for more difficult problems is the hardest part of this section.

2. The most interesting part of this reading is how something as simple as roots, which we learned about in middle school while taking algebra, are being used and applied in this advanced math course. This connection of math principles fascinates me that simple math things can still be used in more advanced courses.

4.3, due on Friday, February 4

1. The most difficult part about this reading was what exactly some of the terms meant because I really had never seen them before or some of them at least I felt were not adequately defined. For example, a nonzero constant polynomial (or a unit) is really confusing to me. I mean, I get what it is, but I had a hard time recognizing them and doing the homework properly because I don't really get what units are in polynomials. The book's definition wasn't good enough. Also, on the homework, we are to prove something about a unique monic associate, and that is in section 4.3, and I have no idea what that is. I used context clues from the problem 1 and its answer, but I still don't understand it enough to use it in a proof.

2. I really like Theorem 4.11 is cool because I love theorems that consistent of 3 or more equivalent statements. Those types of theorems are cool and make writing proofs fun and easy because you can apply whichever equivalent statement fits the proof best. So I like adding another group of equivalent statements to my mathematics knowledge.

4.2, due on Wednesday, February 2

1. This section seems pretty straight forward because it is all definitions and theorems we have seen before, just now they are applied to polynomials. The most difficult part of all of this is how all of them state that f(x) and g(x) need to be in field F. This might be a dumb question and should have probably been cleared up a long time ago, but I do not understand the need or significance or purpose behind the field F and what it means and why it matters. I am not seeing the connection at all.

2. I think it is so interesting how all these definitions and theorems about the gcd and being relatively prime are being applied to polynomials. It is so cool! Also, it is cool how a divisor times any nonzero integer still divides the polynomial. I love unique properties like that!