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.
Thursday, February 17, 2011
Sunday, February 13, 2011
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.
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.
Thursday, February 10, 2011
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.
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.
Monday, February 7, 2011
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.
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.
Sunday, February 6, 2011
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.
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.
Thursday, February 3, 2011
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.
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.
Tuesday, February 1, 2011
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!
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!
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