Reflective:
Like a couple of past readings I enjoyed this one because it was short and sweet. This section was very straightforward and very much like the last section. It is interesting to learn how the division algorithm and the G.C.D can also apply to polynomials. It also makes sense to me how they come to defining a unique G.C.D.
Difficult:
The most difficult part about this reading was understanding how these polynomials apply to a field and also the proof of theorem 4.5. I understood the first half of the proof, but after the first paragraph, the second paragraph was difficult for me to connect to the first. I understand how polynomials can be treated like integers, but it was just difficult to follow the rest of proof 4.5.
Tuesday, January 31, 2012
Thursday, January 26, 2012
Section 4.1 Due January 26, 2012
Difficult:
It was difficult for me to follow the proof on the last two pages. For example I am not sure why they multiplied by a_(n) b_(m)^(-1) x(n-m). It did not make sense to me. I understand long division using polynomials, and I understand how the division algorithm applies to F[x]. But that was all as far as those couple of pages. The rest of the section I understood very well. It is difficult for me to visualize rings and fields and how they relate to this material, but I still understood most of it. Everything was very straightforward for this section. But probably the most difficult part, like I said earlier, was the Division Algorithm in F[x].
Reflective:
I have never thought of x being not any particular number but just being treated like a number. The analogy using pi help a lot with that explanation. It was very interesting in this section to try to understand the connection that R and R[x] have in polynomials. It was also interesting to note that the ring R is a sub-ring of the polynomial ring R[x] and yet, if R is an integral domain, then so is R[x]. Usually it seems like it would be the other way around on the latter statement.
It was difficult for me to follow the proof on the last two pages. For example I am not sure why they multiplied by a_(n) b_(m)^(-1) x(n-m). It did not make sense to me. I understand long division using polynomials, and I understand how the division algorithm applies to F[x]. But that was all as far as those couple of pages. The rest of the section I understood very well. It is difficult for me to visualize rings and fields and how they relate to this material, but I still understood most of it. Everything was very straightforward for this section. But probably the most difficult part, like I said earlier, was the Division Algorithm in F[x].
Reflective:
I have never thought of x being not any particular number but just being treated like a number. The analogy using pi help a lot with that explanation. It was very interesting in this section to try to understand the connection that R and R[x] have in polynomials. It was also interesting to note that the ring R is a sub-ring of the polynomial ring R[x] and yet, if R is an integral domain, then so is R[x]. Usually it seems like it would be the other way around on the latter statement.
Tuesday, January 24, 2012
Section 3.3, Due on January 23, 2012
Difficult:
As usual with these sections I understand most of the first half of the section and then the last half is the most difficult part. I had difficulty understanding all of theorem 3.12 and it's following proof. I was able to understand what a homomorphism is and I feel like I was able to read through the proofs and understand the picture. But it is difficult for me to get a deeper understanding of it so that I can feel more confident.
Reflective:
In general it is interesting to me how math can define one thing and relate it on so many levels to the rest of mathematics. It really helps me to see why they call math a language; because it is so intricate and unique in its patters and how these pattern's connect to the rest of the language.
As usual with these sections I understand most of the first half of the section and then the last half is the most difficult part. I had difficulty understanding all of theorem 3.12 and it's following proof. I was able to understand what a homomorphism is and I feel like I was able to read through the proofs and understand the picture. But it is difficult for me to get a deeper understanding of it so that I can feel more confident.
Reflective:
In general it is interesting to me how math can define one thing and relate it on so many levels to the rest of mathematics. It really helps me to see why they call math a language; because it is so intricate and unique in its patters and how these pattern's connect to the rest of the language.
Thursday, January 19, 2012
Section 3.2 Due January 19, 2012
Difficult:
I followed the first half of the section well, but as it got more complicated it became more difficult to understand. Plus I am still having troubles understanding exactly what a ring is. What is most difficult to me at this point is keep track of the terminology. I do not remember what it means for a ring to be an integral domain. But for the most part I am able to follow the proofs and understand the overall idea of what is happening. Also, a big difficulty is connecting what is being taught now with the axioms. There are many times when the book references an axiom and I do not know what it is talking about. I need to make sure I memorize them.
Reflective:
I find it fascinating that there is the whole other domain it seems that works almost just like the integers, but in a different way. As I was reading through the proofs it made more sense to my why a ring could behave like the integers. It also continues to fascinate me that everything stays so consistent. Math is one, if not the most consistent fields of study that exists I think. I enjoying reading material when it finally makes complete mathematical sense in my head. Some of these proofs were like that.
I followed the first half of the section well, but as it got more complicated it became more difficult to understand. Plus I am still having troubles understanding exactly what a ring is. What is most difficult to me at this point is keep track of the terminology. I do not remember what it means for a ring to be an integral domain. But for the most part I am able to follow the proofs and understand the overall idea of what is happening. Also, a big difficulty is connecting what is being taught now with the axioms. There are many times when the book references an axiom and I do not know what it is talking about. I need to make sure I memorize them.
Reflective:
I find it fascinating that there is the whole other domain it seems that works almost just like the integers, but in a different way. As I was reading through the proofs it made more sense to my why a ring could behave like the integers. It also continues to fascinate me that everything stays so consistent. Math is one, if not the most consistent fields of study that exists I think. I enjoying reading material when it finally makes complete mathematical sense in my head. Some of these proofs were like that.
Tuesday, January 17, 2012
Section 3.1, Due January 17, 2012
Difficult:
I think I understand the overall idea of rings, but I am still having difficulty completely understanding how to prove that something is a ring. Particularly axiom 4 and 5 of the definition. I understand the concept that there needs to be a solution to a+0=a=0. But the subtext in the text book throws me off. I do not completely understand that.
Reflective:
I like the overall concept of abstract algebra. From the book I liked the sentence that said that "By 'abstracting' the common core of essential features, we can develop a general theory that includes as special cases Z, Z (sub n) and other familiar systems. I like the idea of simply relating systems by what they have in common to make the process more simple. I am excited to learn more in abstract algebra, I just hope that it doesn't get too abstract. :)
I think I understand the overall idea of rings, but I am still having difficulty completely understanding how to prove that something is a ring. Particularly axiom 4 and 5 of the definition. I understand the concept that there needs to be a solution to a+0=a=0. But the subtext in the text book throws me off. I do not completely understand that.
Reflective:
I like the overall concept of abstract algebra. From the book I liked the sentence that said that "By 'abstracting' the common core of essential features, we can develop a general theory that includes as special cases Z, Z (sub n) and other familiar systems. I like the idea of simply relating systems by what they have in common to make the process more simple. I am excited to learn more in abstract algebra, I just hope that it doesn't get too abstract. :)
Wednesday, January 11, 2012
Section 2.2, Due January 11, 2012
Difficult Parts:
Most of this section was very strait forward for me and a simply review of what I have already learned in Math 290. But the most difficult part about this material has always been understanding that other equivalence classes are the same as other equivalence classes and so on.
Reflective:
Surprisingly, the part that I found to be the most difficult to understand was also the part that I think is most interesting about this section. They way the classes work together with regular algebra and the fact that there are only n-1 distinct equivalence classes, fascinates me.
Most of this section was very strait forward for me and a simply review of what I have already learned in Math 290. But the most difficult part about this material has always been understanding that other equivalence classes are the same as other equivalence classes and so on.
Reflective:
Surprisingly, the part that I found to be the most difficult to understand was also the part that I think is most interesting about this section. They way the classes work together with regular algebra and the fact that there are only n-1 distinct equivalence classes, fascinates me.
Sunday, January 8, 2012
Section 2.1, Due on January 8, 2012
Difficult:
Overall, not much was difficult for me in this section. But if I had to label one part as being the most difficult it was when they made the connection between congruence and the Division Algorithm. Specifically, Corollary 2.5 was more difficult for me to grasp. Especially part (1). The statement that if any integer is divided by n, the congruence class of that integer and its remainder will be equal. This was difficult for me because the part of the Corollary seemed to simple, basic and general. I found myself thinking, "How is that possible for ANY integer?" But then as I read the proof, it made more obvious sense to me.
Reflective:
I enjoyed reading this section. The way that the book represented equality really stuck out to me. Their definitions of reflexive, symmetric and transitive, though familiar to me were presented in a slightly more simplistic way then I remember. I think I had always associated those three characteristics with equivalence relations and not just with basic numbers. As I read through the definitions carefully, it made deeper sense to me this time than it had before.
Overall, not much was difficult for me in this section. But if I had to label one part as being the most difficult it was when they made the connection between congruence and the Division Algorithm. Specifically, Corollary 2.5 was more difficult for me to grasp. Especially part (1). The statement that if any integer is divided by n, the congruence class of that integer and its remainder will be equal. This was difficult for me because the part of the Corollary seemed to simple, basic and general. I found myself thinking, "How is that possible for ANY integer?" But then as I read the proof, it made more obvious sense to me.
Reflective:
I enjoyed reading this section. The way that the book represented equality really stuck out to me. Their definitions of reflexive, symmetric and transitive, though familiar to me were presented in a slightly more simplistic way then I remember. I think I had always associated those three characteristics with equivalence relations and not just with basic numbers. As I read through the definitions carefully, it made deeper sense to me this time than it had before.
Thursday, January 5, 2012
Sections 1.1-1.3, Due on January 5, 2012
1. Difficult
I find it most difficult with material likes this to focus on the overall idea that is being presented in the text and proofs. I get so caught up and easily confused by the letters, equations and words that many times it seems to go in one ear and out the other. I know that right now most of what we are learning is mostly review, but even with material where I know the answers and I have already understood the overall concept I still get caught up on the details and it is difficult for me to completely judge in my head what the book is actually trying to say.
It is also difficult for me to completely see or judge why or how a proof actually proves something. Often times when I read a theorem, especially when it is overly simple, I think, oh that is such a basic fact, how on earth could you prove it? And then the books proof's strategy is a process or something that I wouldn't have even thought to come up with for the answer.
2. Reflective
I really enjoy reading about simple basic facts and why they are true. Of course sometimes I loose focus because the topic gets a little deeper than I would prefer, but most of the time it is great! It gives me a sense of security about some basic math facts that I know to be true. It is also fascinating to how all of the parts are related to one another. For example, it did not occur to me that the gcd of two numbers can obviously be written as a linear combination of the two numbers. I think I always new that, but I never really acknowledged it until know. Now that makes me curious, because I know that we used linear combinations all the time in Math 313. So I am interested to see how this course uses this information and what we have learned in Math 313 to relate it to abstract algebra.
I find it most difficult with material likes this to focus on the overall idea that is being presented in the text and proofs. I get so caught up and easily confused by the letters, equations and words that many times it seems to go in one ear and out the other. I know that right now most of what we are learning is mostly review, but even with material where I know the answers and I have already understood the overall concept I still get caught up on the details and it is difficult for me to completely judge in my head what the book is actually trying to say.
It is also difficult for me to completely see or judge why or how a proof actually proves something. Often times when I read a theorem, especially when it is overly simple, I think, oh that is such a basic fact, how on earth could you prove it? And then the books proof's strategy is a process or something that I wouldn't have even thought to come up with for the answer.
2. Reflective
I really enjoy reading about simple basic facts and why they are true. Of course sometimes I loose focus because the topic gets a little deeper than I would prefer, but most of the time it is great! It gives me a sense of security about some basic math facts that I know to be true. It is also fascinating to how all of the parts are related to one another. For example, it did not occur to me that the gcd of two numbers can obviously be written as a linear combination of the two numbers. I think I always new that, but I never really acknowledged it until know. Now that makes me curious, because I know that we used linear combinations all the time in Math 313. So I am interested to see how this course uses this information and what we have learned in Math 313 to relate it to abstract algebra.
Introduction, Due on January 5, 2012
Hello! This is Abbylynn Payne, and I am taking Abstract Algebra, Math 371 at Brigham Young University from Dr. Jenkins this winter semester 2012. This blog is part of an assignment for that class. It will include, for the most part, posts describing each section of our textbook. Here is a brief introduction of myself as Dr. Jenkins has required:
-I am a Senior at BYU and my major is Math Education.
-After calculus I have taken Math 290 and Math 313.
-I am taking this class because it is a requirement for my major, and because I enjoy math. Specifically, algebra was one of my favorite courses in school, so I am excited to expand my learning of algebra by taking abstract algebra.
-Last semester I had two Math professors who I thought were some of the most effective teachers I have ever had. In my opinion, they were most effective because they were both VERY organized, and one of them was VERY good at explaining things. The reason he was particularly effective was because he REALLY simplified what he said and HOW he said it. Some teachers get excited and talk quickly and in a complicated way, but this professor spoke very slowly and clearly. That helped him to be the best math teacher I have every had.
-Something interesting about me is that I was on the BYU Ballroom Dance Team for three years at BYU and now I have been given the wonderful opportunity to teach ballroom at Pleasant Grove Junior High, part time.
-One time that works really well for me to get help with homework is MWF from 1-2pm.
-I am a Senior at BYU and my major is Math Education.
-After calculus I have taken Math 290 and Math 313.
-I am taking this class because it is a requirement for my major, and because I enjoy math. Specifically, algebra was one of my favorite courses in school, so I am excited to expand my learning of algebra by taking abstract algebra.
-Last semester I had two Math professors who I thought were some of the most effective teachers I have ever had. In my opinion, they were most effective because they were both VERY organized, and one of them was VERY good at explaining things. The reason he was particularly effective was because he REALLY simplified what he said and HOW he said it. Some teachers get excited and talk quickly and in a complicated way, but this professor spoke very slowly and clearly. That helped him to be the best math teacher I have every had.
-Something interesting about me is that I was on the BYU Ballroom Dance Team for three years at BYU and now I have been given the wonderful opportunity to teach ballroom at Pleasant Grove Junior High, part time.
-One time that works really well for me to get help with homework is MWF from 1-2pm.
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