I feel like the most important things we have covered in this semester are the big topics. We have focused a lot on rings, ideal, and groups. I feel that these will be the main topics covered on the final, but the final will probably have more specific questions associated with them. I feel like through out the semester, most of the things we covered revolved around these topics.
The most difficult thing for me this semester has been quotient groups, rings, fields, and anything basically involving cosets. I just feel like I panic when I see R/I or G/N. I also feel like I never completely understood normal groups. But mostly its quotient stuff that I am really uncertain about. I would really like to see a problem involving quotient rings or groups. Something involving a ring modulo an ideal would be helpful. I just don't know what to do with these things when I see them in a problem.
In the future, I think this class will help me form more logical arguments. I have learned to not freak out when I see the word proof. Honestly it has given me more confidence in proving things, (especially since theory of analysis shot down any confidence I had from 290). For the most part, I think this class has helped me understand modular arithmetic really well. I didn't have much exposure to this before, so its is nice to feel confident in my ability to do modular arithmetic and actually understand what is going on.
Tuesday, April 12, 2011
Saturday, April 9, 2011
8.3, Due April 11
Difficult: I had a hard time understanding what a Sylow p-group is. I don't understand how to find these sylow p-groups and why they are even important. First of all, I don't even know what sylow means. How are sylow p-groups different than regular ones. Even the example following the definition didn't help me understand this concept any better. I don't know how to use the definition to help me use and find the sylow p-groups.
Connections: I felt like the first sylow theorem sounded really familiar. It seemed to me like Lagrange's theorem had some kind of relation to it. This theorem made sense to me, and it seemed that logically it just followed from Lagrange's Theorem. I would understand how to apply this theorem since it sounds a lot like previous theorems we have used in this class.
Connections: I felt like the first sylow theorem sounded really familiar. It seemed to me like Lagrange's theorem had some kind of relation to it. This theorem made sense to me, and it seemed that logically it just followed from Lagrange's Theorem. I would understand how to apply this theorem since it sounds a lot like previous theorems we have used in this class.
Thursday, April 7, 2011
8.2, Due April 8
Difficult: In the beginning of the section, there was notation G(p) for the set of elements in G whose order is some prime power of p. This notation was confusing to me, and I didn't really understand what it was talking about. Even the example immediately following was of no help to me. Then the following lemmas and theorems all included this notation so I had a hard time following what they were even stating. So following the proof was even more difficult. I felt like this whole section pretty much followed this idea of prime powers and and p-groups, so I felt confused about what this section was stating.
Connections: This section uses a lot of things which we have visited previously in this class. For example, the notion of the orders of elements in a group being prime powers. We have spent lots of time discussing the orders of the elements of a group, and also discussing primes. We have also recently talked about cyclic groups and this is very important in the statement of the Fundamental Theorem of Finite abelian groups. So there are a lot of things in this section that we have visited before, but I feel like the prime power order is the thing that is most confusing.
Connections: This section uses a lot of things which we have visited previously in this class. For example, the notion of the orders of elements in a group being prime powers. We have spent lots of time discussing the orders of the elements of a group, and also discussing primes. We have also recently talked about cyclic groups and this is very important in the statement of the Fundamental Theorem of Finite abelian groups. So there are a lot of things in this section that we have visited before, but I feel like the prime power order is the thing that is most confusing.
Tuesday, April 5, 2011
8.1, Due April 6
Difficult: I was doing pretty good with the first part of this section, and then I started getting confused by the theorems. For example, theorem 8.1. I did not understand what this means at all. Also, lemma 8.2 was confusing to me as well. I didn't understand how this holds true. I might still be confused by normal subgroups and how they apply in proofs. I am trying to figure our how to follow the proof of 8.1, but I just don't understand all the subscripts and notation. I also don't get how to find what MxN is isomorphic to like the 2nd example of the section.
Connections: I recognize seeing the cartesian products in other sections of this text. I understand what to do with them and how to find the cartesian products, so using them in proofs should be pretty easy for me, but like I said above some of the notation is getting a little confusing for me. I feel like the cartesian products are pretty computational and are easy to work with and to find the elements of a cartesian product.
Connections: I recognize seeing the cartesian products in other sections of this text. I understand what to do with them and how to find the cartesian products, so using them in proofs should be pretty easy for me, but like I said above some of the notation is getting a little confusing for me. I feel like the cartesian products are pretty computational and are easy to work with and to find the elements of a cartesian product.
Friday, April 1, 2011
7.10, Due April 4
Difficult: As I was reading in this section, I got a little lost in the proof of 7.52 where it says that N is a normal subgroup of An. What does this subgroup look like? I think that it is just a some of the elements of Sn. So is it just a set of different cycles of An? I think I am also having a hard time figuring out what An looks like. How many elements are there? Is there a way to find this out by having a certain length of a cycle or knowing what symmetric group you are working in? Also, the proof of 7.52 is so long that I just got way lost in all of the notation and didn’t really understand what was going on.
Connections: Now that I feel like I better understand the new notation with symmetric groups, I feel like I can understand the notation used in the proofs involving cycles. I like that the cycles are pretty easy to compute the “multiplication” of them. I think that knowing this information helps me be able to follow the composition of the cycles when it is done in the proofs.
Thursday, March 31, 2011
7.9, Due April 1
I decided to write this blog about unicorns today instead of Abstract Algebra.. Enjoy!!!
APRIL FOOLS!!!
For reals though, here is the real blog information
Difficult: I felt that they new notation for the symmetric groups was really confusing and I feel like it isn't useful at all. What is the point of trying to write it this way, I have to think about it way more than I did when we just wrote it all out. So cycles are also confusing to me since I don't understand the notation. I got really confused about the notation because I felt like I understood some of it, but the notation on the middle of pg 231 didn't make any sense to me. So for the rest of the section, I was so confused because I couldn't make sense of cycles and the notation. So transpositions and alternating groups was really confusing to me.
Connections: The only connections I can really make with this section is that we have seen the symmetric groups before in other sections of chapter 7. I like them because they are good examples of non-abelian groups with different orders. However, they were kind of tedious to write them all out, so maybe once I can understand the new notation I will like them better and be able to understand how to have more applications of them.
APRIL FOOLS!!!
For reals though, here is the real blog information
Difficult: I felt that they new notation for the symmetric groups was really confusing and I feel like it isn't useful at all. What is the point of trying to write it this way, I have to think about it way more than I did when we just wrote it all out. So cycles are also confusing to me since I don't understand the notation. I got really confused about the notation because I felt like I understood some of it, but the notation on the middle of pg 231 didn't make any sense to me. So for the rest of the section, I was so confused because I couldn't make sense of cycles and the notation. So transpositions and alternating groups was really confusing to me.
Connections: The only connections I can really make with this section is that we have seen the symmetric groups before in other sections of chapter 7. I like them because they are good examples of non-abelian groups with different orders. However, they were kind of tedious to write them all out, so maybe once I can understand the new notation I will like them better and be able to understand how to have more applications of them.
Monday, March 28, 2011
7.8, Due March 30
Difficult: So in the first part of this section, the ideas are pretty parallel to those of rings. I understand homomorphisms, but the hardest part for me is finding the map f that is a homomorphism. It is easier for me to prove, but is there some kind of way to know what the map is? In all of the examples in the book, they choose their map, but this is something that is hard for me to do. Also, the idea of simple groups makes sense to me, but I'm just not sure how to go about testing if a group is simple. Is there a general way to do it? Also, I kind of got lost in the explanation of why simple groups are important. I didn't understand the point of them, I think I might have just gotten lost in the notation.
Connections: The First Isomorphism Theorem for groups is very similar to that for rings. I feel very confident in my ability to prove this theorem for rings, and I feel that I understand it pretty well. As a result, I think that I would be able to do this proof for groups as well. It follows the same general idea, but some of the notation is changed and some definitions would look different. Again, I am left wondering how they came up with the map for phi? How did they know to pick phi(Ka)=f(a)?
Connections: The First Isomorphism Theorem for groups is very similar to that for rings. I feel very confident in my ability to prove this theorem for rings, and I feel that I understand it pretty well. As a result, I think that I would be able to do this proof for groups as well. It follows the same general idea, but some of the notation is changed and some definitions would look different. Again, I am left wondering how they came up with the map for phi? How did they know to pick phi(Ka)=f(a)?
7.7, Due March 28
Difficult: Just like with quotient rings with ideals, I don't understand how to determine if something is a quotient ring, and how to find the elements of it. Also, at the end of the section it was talking about the structure of the quotient groups. I don't understand why this is important and when it would be useful. I think the hardest part of this section was just being able to find the quotient group and know why it was useful.
Connections: The theorems and ideas associated with quotient groups is pretty similar to those of quotient rings. I feel like these are a lot alike. However, as I said before, quotient rings were difficult for me to understand, so I also feel like quotient groups will be hard to understand. Also, I noticed that quotient groups were given a new name as well, factor groups. Why are they called that? Is there a property about them that has to do with factoring?
Connections: The theorems and ideas associated with quotient groups is pretty similar to those of quotient rings. I feel like these are a lot alike. However, as I said before, quotient rings were difficult for me to understand, so I also feel like quotient groups will be hard to understand. Also, I noticed that quotient groups were given a new name as well, factor groups. Why are they called that? Is there a property about them that has to do with factoring?
Thursday, March 24, 2011
7.6 Part 2, Due March 25
So I forgot to post on the first part of section 7.6... but here is the second stuff
Difficult: For me, the most difficult thing about normal subgroups is just realizing that aN=Na, not that an=na. The It is hard for me to realize that the sets just have to be equal, not that every element has to commute. Another thing that I find difficult is just being able to find whether aNa^(-1) is a subset of N. I just have a hard time remembering how to find if something is a subset... its just something that I need to review.
Connections: Theorem 7.33 is similar to most of the congruence theorems that we have seen this semester about multiplying two things that are congruent modulo something. However, this one is a little different because the two things are congruent modulo a normal subgroup. This one seems to have a little bit more of a restriction. I also noticed that there wasn't a theorem for adding two things that are congruent modulo a normal subgroup. Is this not allowed?
Difficult: For me, the most difficult thing about normal subgroups is just realizing that aN=Na, not that an=na. The It is hard for me to realize that the sets just have to be equal, not that every element has to commute. Another thing that I find difficult is just being able to find whether aNa^(-1) is a subset of N. I just have a hard time remembering how to find if something is a subset... its just something that I need to review.
Connections: Theorem 7.33 is similar to most of the congruence theorems that we have seen this semester about multiplying two things that are congruent modulo something. However, this one is a little different because the two things are congruent modulo a normal subgroup. This one seems to have a little bit more of a restriction. I also noticed that there wasn't a theorem for adding two things that are congruent modulo a normal subgroup. Is this not allowed?
Sunday, March 20, 2011
Midterm #2 Post, Due March 21
For this section of the book, I think that Lagrange's Theorem, Cayley's Theorem, and the first Isomorphism theorem are pretty important. Usually when theorems are named they are a big deal. I also think that other theorems will be really important too just because it is useful information to know in helping prove other things. For example, I feel like the theorems that give easier or faster ways to determine if something is a sub ring, or subgroup are really helpful. Also, the theorems that just are simple things to know to be able to use in other proofs, for example inverses are unique, or cancellation holds, or the identity is unique.
I expect to see questions about proving if something is an ideal, or a group. But I could also see things like find a maximal ideal, or prove something about the kernel, or show that this group is cyclic. I think there will also be questions about finding the cosets of an ideal or group. Also, like the last exam, I think there will be questions asking for examples of groups, ideals, subgroups, etc.
Something that I would like to discuss before the exam is that I really want to go over the main differences of the main topics we have covered in this unit. Another thing that I really struggle with is maximal ideals, cosets, and cyclic-generated groups. So I would really like to see examples of these things so that maybe some of my confusion could be cleared up.
I expect to see questions about proving if something is an ideal, or a group. But I could also see things like find a maximal ideal, or prove something about the kernel, or show that this group is cyclic. I think there will also be questions about finding the cosets of an ideal or group. Also, like the last exam, I think there will be questions asking for examples of groups, ideals, subgroups, etc.
Something that I would like to discuss before the exam is that I really want to go over the main differences of the main topics we have covered in this unit. Another thing that I really struggle with is maximal ideals, cosets, and cyclic-generated groups. So I would really like to see examples of these things so that maybe some of my confusion could be cleared up.
Thursday, March 17, 2011
7.5 Part 2, Due March 18
Difficult: For me, I have found that finding and proving groups are isomorphic is rather difficult. I feel like the only way to do this is to create some map, f, and then write out the multiplication/operation tables to show that they are the isomorphic. However, I feel like this could be an insufficient way of proving this, but I have no clue how to show this any other way. So from this section, I still just don't know how to show something is isomorphic with out multiplication tables. And beyond this section, I realized that I am just super confused about cyclic groups and how to know if a group is cyclic and how to find the order of a group as well as its elements. This is probably something I should know, but as I was trying to do the homework for the last section, I found that these concepts were causing me grief.
Connections: I guess the only connections I can make with this section is that the integers modulo a number can be isomorphic to other things, ie rings, groups, etc. So this was something I felt like I understood, however, as I mentioned above, I still find it really hard to find the isomorphic map and how to find if that is indeed a homomorphism. However, I still feel like the definition of an isomorphism is something I have seen before, but I find that I have a harder time showing isomorphisms for groups.
Connections: I guess the only connections I can make with this section is that the integers modulo a number can be isomorphic to other things, ie rings, groups, etc. So this was something I felt like I understood, however, as I mentioned above, I still find it really hard to find the isomorphic map and how to find if that is indeed a homomorphism. However, I still feel like the definition of an isomorphism is something I have seen before, but I find that I have a harder time showing isomorphisms for groups.
Tuesday, March 15, 2011
Vitaly Bergelson, Make-Up blog
Difficult: In his talk, Vitaly Bergelson talked about Poincare's Recurrence Theorem which had something to do with volume preserving transformations. I had absolutely no idea of what he was talking about. I tried to understand it, but I think I might have just gotten lost in the notation. He asked if we had any questions, but I was so confused so I didn't even know what to ask. He also talked about upper density functions. I didn't even know to what this is and how to find them.
Connections: The only connections I could find with this talk was about finding volume. This reminded me of multivariable calculus since I found volume of the three dimensional figures. Also, I felt like in linear algebra we talked about n-dimensional stuff, but I'm not sure exactly what we did with it. I couldn't really find many connections with this since I felt so confused.
Connections: The only connections I could find with this talk was about finding volume. This reminded me of multivariable calculus since I found volume of the three dimensional figures. Also, I felt like in linear algebra we talked about n-dimensional stuff, but I'm not sure exactly what we did with it. I couldn't really find many connections with this since I felt so confused.
Monday, March 14, 2011
7.5 Part 1, Due on March 16
Difficult: I think that the definition of right cosets is kind of confusing. However, it reminds me of some things we did with ideals. I think that the notation is just confusing to me. Also, I don't really understand what makes it a right cosets... what is a left coset? Thus, I didn't really understand theorem 7.25 because I don't understand even what the union of all the cosets would be.
Connections: In this section, I think I understand what it means for two elements of a group to be congruent modulo a subgroup. It is pretty similar to how it has been defined previously with rings, and ideals. I also understand how congruence modulo a subgroup is also an equivalence relation just as congruence has been in previous sections. Like I said before, I think I understood congruence, but the right coset thing is still a little weird to me, and also I am left wondering why do we care about the right coset anyway? What is so important about the right coset??
Connections: In this section, I think I understand what it means for two elements of a group to be congruent modulo a subgroup. It is pretty similar to how it has been defined previously with rings, and ideals. I also understand how congruence modulo a subgroup is also an equivalence relation just as congruence has been in previous sections. Like I said before, I think I understood congruence, but the right coset thing is still a little weird to me, and also I am left wondering why do we care about the right coset anyway? What is so important about the right coset??
Thursday, March 10, 2011
7.3, Due March 11
Difficult: I was doing pretty well with this section until I got to the section on cyclic groups. I did not understand this section of the reading at all. The notation was also kind of confusing to me. I just didn't understand what <a> means. The set of all the powers of "a" confused me because I thought that one of the powers of a should eventually equal the identity. Is this wrong? Or is that only when there is finite order? I just got confused by what cyclic groups really are I guess.
Connections: I like the idea of subgroups though. They seem pretty simple to me, especially after dealing with subrings. The theorems that go with subgroups, at least in the beginning of the section, seem to be pretty simple. The theorems seem pretty straight forward to compute to show whether a subset of a group is a subgroup. These theorems were easy for me to follow so I should be able to use them in the proofs for this section.
Connections: I like the idea of subgroups though. They seem pretty simple to me, especially after dealing with subrings. The theorems that go with subgroups, at least in the beginning of the section, seem to be pretty simple. The theorems seem pretty straight forward to compute to show whether a subset of a group is a subgroup. These theorems were easy for me to follow so I should be able to use them in the proofs for this section.
Tuesday, March 8, 2011
7.2, Due on March 9
Difficult: I had a hard time understanding the order of an element, and how to find it. The more difficult part for me though, was to find out whether or not an element has distinct powers by looking at the elements in a group. The example on the top of page 177 was difficult for me. I understood what it was talking about, but I don't think I would know to how figure out the powers on the exponents on my own. I also thought it was difficult to find out if an element has finite or infinite order.
Connections: I like in this section that some of the properties of groups are properties that I have used in other parts of mathematics. For example, the inverse of ab is b inverse times a inverse. This is similar to concepts used in linear algebra. Also, some of the properties of exponents hold as well. Since I have been using these concepts for a while, I am more comfortable using them and applying them to new ideas such as groups.
Connections: I like in this section that some of the properties of groups are properties that I have used in other parts of mathematics. For example, the inverse of ab is b inverse times a inverse. This is similar to concepts used in linear algebra. Also, some of the properties of exponents hold as well. Since I have been using these concepts for a while, I am more comfortable using them and applying them to new ideas such as groups.
Sunday, March 6, 2011
7.1 Part 2, Due March 7
Difficult: One thing that I thought was difficult about this section was the dihedral groups. I felt like I understood them, and I understood why they were groups since you can think about the group as a composition of functions. However, the thing I thought that was the most difficult part was trying to figure out the symmetries of the figures. Especially with the equilateral triangle. If I was asked to find the symmetries of a dihedral group of degree n, I wouldn't know how many symmetries there are supposed to be. Is there some way to know how many symmetries there should be in a dihedral group of degree n?
Connections: I like that groups seem to be really similar to rings, but even more simple. I also like that we can look at familiar rings and notice whether or not they can be groups. It helps me get more used to groups since we are looking at rings and other things that we have already considered in this course. At the end of the section, they talked about different groups that can be formed from things that we already have worked with. I like that groups are pretty simple and easy to work with, just like rings again.
Connections: I like that groups seem to be really similar to rings, but even more simple. I also like that we can look at familiar rings and notice whether or not they can be groups. It helps me get more used to groups since we are looking at rings and other things that we have already considered in this course. At the end of the section, they talked about different groups that can be formed from things that we already have worked with. I like that groups are pretty simple and easy to work with, just like rings again.
Thursday, March 3, 2011
7.1 Part 1, Due March 4
Difficult: For me, this section was pretty easy to understand, especially after making it through Chapter 6. However, I still have some questions. I was wondering what binary operations are allowed for groups. I guess I just am confused how to know how this operation is defined, or for that matter, how to define the operation. It seems like groups are a lot like rings, but I'm not quite sure what makes them different from rings. Is it just that only one operation is defined?
Connection: I like that in this section we are going back to something that seems more computational, and therefore a little less abstract. I am probably wrong though. I did like in this section how it talked about the composition of permutations. I like that this is very easy for me to follow and understand because I have been doing composition of functions for quite some time. I am hoping that groups will be a little more straightforward and simple to understand.
Connection: I like that in this section we are going back to something that seems more computational, and therefore a little less abstract. I am probably wrong though. I did like in this section how it talked about the composition of permutations. I like that this is very easy for me to follow and understand because I have been doing composition of functions for quite some time. I am hoping that groups will be a little more straightforward and simple to understand.
Tuesday, March 1, 2011
6.3, Due March 2
Difficult: I had a hard time understanding what maximal ideals are. I think that maybe it is notation that is confusing me, but I just couldn't quite understand what it means for an ideal to be maximal. As I was reading the definition, I also realized that I'm not really sure what it means for two ideals to be equal. So I don't know how to go about proving something is maximal. Mostly because I'm not sure how to prove that two ideals are equal.
Connections: I like dealing with prime numbers, or concepts having to do with primes. I think that things that are prime have very nice properties and are really nice to work with. Some of the examples were pretty simple, so I think for the most part I understood them. However, sometimes I think I understand things and I try to do the homework and I'm not quite sure how to apply things. So hopefully applying the concepts of prime ideals will be something I will be able to work with, and I will be able to understand how to use information about ideals being prime help me out in some of the proofs.
Connections: I like dealing with prime numbers, or concepts having to do with primes. I think that things that are prime have very nice properties and are really nice to work with. Some of the examples were pretty simple, so I think for the most part I understood them. However, sometimes I think I understand things and I try to do the homework and I'm not quite sure how to apply things. So hopefully applying the concepts of prime ideals will be something I will be able to work with, and I will be able to understand how to use information about ideals being prime help me out in some of the proofs.
Sunday, February 27, 2011
6.2 Part 2, Due February 28
Difficult: I don't really understand the first isomorphic theorem. I don't understand how to use and apply it to what we are working with. The examples on page 150 confused me even further. I think it will be difficult to find a function that is a surjective homomorphism, and to find the kernel, and use this information to find out the structure of quotient rings. I think mostly, I just don't know what this theorem even means! So knowing when it is applicable is going to be quite a challenge.
Connections: I like that ideals are related to homomorphisms as well. I like working with homomorphisms and isomorphisms mostly because I feel that the proofs with them are very computational... which I really like. I like that for most of these theorems, one of the givens is that it is a homomorphism. This makes going through the proofs fairly computational when we get to use this information. It is simple that homomorphisms are defined by two things, which are easy to show in a proof.
Connections: I like that ideals are related to homomorphisms as well. I like working with homomorphisms and isomorphisms mostly because I feel that the proofs with them are very computational... which I really like. I like that for most of these theorems, one of the givens is that it is a homomorphism. This makes going through the proofs fairly computational when we get to use this information. It is simple that homomorphisms are defined by two things, which are easy to show in a proof.
Thursday, February 24, 2011
6.2 part 1, Due February 25
Difficult: I feel like the most difficult part of this section was still just understanding cosets. I felt like I understood pretty well how to add and multiply cosets, but I still don't get how you find the cosets of these rings. I feel like maybe the notation is a little foreign to me. I don't quite know how to find all of the cosets of the form a+I, so adding and multiplying will be quite difficult if I don't even know what the cosets are.
Connections: I thought that the multiplication and addition tables made sense to me. I feel like we have done a lot of addition and multiplication tables, so I think that once I know the cosets of the ring, I will be able to make the multiplication and addition tables as long as it is not an infinite set of cosets. However, as I mentioned above, I'm not convinced that I will be able to find the cosets... which is a slight problem if I need to try to add and multiply them together.
Connections: I thought that the multiplication and addition tables made sense to me. I feel like we have done a lot of addition and multiplication tables, so I think that once I know the cosets of the ring, I will be able to make the multiplication and addition tables as long as it is not an infinite set of cosets. However, as I mentioned above, I'm not convinced that I will be able to find the cosets... which is a slight problem if I need to try to add and multiply them together.
Tuesday, February 22, 2011
6.1 part 2, Due February 23
Difficult: For me, the hardest part of this section was finding the congruence classes. Just like with polynomials, finding congruence classes is very difficult. I have a hard time deciding how many congruence classes there are, and what they even are! I think however, that this section is more difficult than polynomials to determine the congruence classes. I don't quite understand the notation well enough to be able to find what the congruence classes are. Also, I don't really get when a (left) coset is either. How is this different than a congruence class, or is it the same?
Connections: I like that congruence is similar to the congruence that we have been dealing with in the polynomials, or in the integers. I like consistency in definitions, and so I like that this definition is very familiar. It is easier for me to remember when it follows the same pattern as the other kinds of congruence. I also like that theorems 6.4 and 6.5 are similar to the theorems in the other sections about how congruence is reflexive, symmetric, and transitive and that addition and multiplication can follow in congruence. These definitions are definitely something I have seem before, so its nice to have something familiar attached to the new ideas of ideals.
Connections: I like that congruence is similar to the congruence that we have been dealing with in the polynomials, or in the integers. I like consistency in definitions, and so I like that this definition is very familiar. It is easier for me to remember when it follows the same pattern as the other kinds of congruence. I also like that theorems 6.4 and 6.5 are similar to the theorems in the other sections about how congruence is reflexive, symmetric, and transitive and that addition and multiplication can follow in congruence. These definitions are definitely something I have seem before, so its nice to have something familiar attached to the new ideas of ideals.
Monday, February 21, 2011
6.1 part 1, Due February 22
Difficult: I felt like in this section I understood the idea of ideals, until it started talking about the different kinds of ideals. I really liked theorem 6.1, and I thought it was pretty straight-forward. However, the principal ideals, and finitely generated ideals were a little confusing to me. I wasn't quite sure what the difference was between them, and how they differed from just being an ideal. I found it difficult to distinguish between them and understand why they were important.
Connections: As I mentioned before, I liked the idea of the ideals in the beginning. I felt like it sounded very familiar to stuff we have already learned this semester. Theorem 6.1 reminded me a lot of a subring! The properties to be an ideal seem to me to be a lot like the properties to be a subring. I like that the properties of being an ideal are that they are closed under subtraction and multiplication, even though it is a little different from a subring since in the multiplication, one of the elements need not be in the set I, but the product of an element from within the set and outside the set need only be in I.
Connections: As I mentioned before, I liked the idea of the ideals in the beginning. I felt like it sounded very familiar to stuff we have already learned this semester. Theorem 6.1 reminded me a lot of a subring! The properties to be an ideal seem to me to be a lot like the properties to be a subring. I like that the properties of being an ideal are that they are closed under subtraction and multiplication, even though it is a little different from a subring since in the multiplication, one of the elements need not be in the set I, but the product of an element from within the set and outside the set need only be in I.
Thursday, February 17, 2011
5.3, Due February 18
Difficult: The thing I found most difficult in this section was the idea of an extension field. I don't really get the point of this. So it makes irreducible polynomials have roots in the extension field, but I couldn't figure out what those roots are, or why it matters. I was also didn't understand how the extension field can help us define complex numbers either. Basically, I just don't understand why extension fields are important, and what it helps us do.
Connections: So I feel like I am starting to get the hang of working with polynomials modulo p(x), but it is pretty different from modular arithmetic with integers. I also like that I am getting more practice with irreducible polynomials, and finding them. I like using the root test to find or verify that a polynomial is reducible or irreducible. This is a simpler way for me than trying to find all of the reducible ones and picking one that is not in that list. So it is nice to work more with irreducible polynomials in this section so that I can get more comfortable with them.
Connections: So I feel like I am starting to get the hang of working with polynomials modulo p(x), but it is pretty different from modular arithmetic with integers. I also like that I am getting more practice with irreducible polynomials, and finding them. I like using the root test to find or verify that a polynomial is reducible or irreducible. This is a simpler way for me than trying to find all of the reducible ones and picking one that is not in that list. So it is nice to work more with irreducible polynomials in this section so that I can get more comfortable with them.
Saturday, February 12, 2011
5.1, Due on February 14
Difficult:
For me, the most difficult part of this section was when it talked about the congruence classes. I think this would be so hard to find the congruence classes, and determine how many congruence classes there are. This section is going to be more difficult than congruence classes of integers because I think dividing polynomials is way harder! I am going to have a hard time finding congruence classes, and even knowing what things are of the same congruence class.
Connections:
I like that this section pretty much mirrors the section on congruence classes for integers. It follows the same kinds of properties and definitions as it did in the other section, but this time the dividing part will be much more difficult. It will be harder to recognize when something is in the same congruence classes because sometimes it is harder to see what is going on with polynomials. Hopefully my understanding of congruence classes for integers will help me understand it better for polynomials.
For me, the most difficult part of this section was when it talked about the congruence classes. I think this would be so hard to find the congruence classes, and determine how many congruence classes there are. This section is going to be more difficult than congruence classes of integers because I think dividing polynomials is way harder! I am going to have a hard time finding congruence classes, and even knowing what things are of the same congruence class.
Connections:
I like that this section pretty much mirrors the section on congruence classes for integers. It follows the same kinds of properties and definitions as it did in the other section, but this time the dividing part will be much more difficult. It will be harder to recognize when something is in the same congruence classes because sometimes it is harder to see what is going on with polynomials. Hopefully my understanding of congruence classes for integers will help me understand it better for polynomials.
Thursday, February 10, 2011
9.4, Due on February 11
Difficult:
I had a hard time just plain understanding what this section was talking about. I got lost at the relation ~, and I never really found my way back. I found it difficult to understand how they were deciding to define multiplication and addition, and I think this is because I was confused what they were doing with quotients and how they were turning them into an ordered pair and adding them. So much of this section built upon the beginning that I was confused pretty much throughout the whole section because I didn't even understand the beginning.
Connections:
I feel like since I know what an integral domain and a field are, I should be able to better understand this section by connecting the new concepts involving the quotients and applying them to integral domains and fields. I feel pretty comfortable with the definitions of these, so I think that I can use this knowledge to help me make sense of the new application using quotients.
I had a hard time just plain understanding what this section was talking about. I got lost at the relation ~, and I never really found my way back. I found it difficult to understand how they were deciding to define multiplication and addition, and I think this is because I was confused what they were doing with quotients and how they were turning them into an ordered pair and adding them. So much of this section built upon the beginning that I was confused pretty much throughout the whole section because I didn't even understand the beginning.
Connections:
I feel like since I know what an integral domain and a field are, I should be able to better understand this section by connecting the new concepts involving the quotients and applying them to integral domains and fields. I feel pretty comfortable with the definitions of these, so I think that I can use this knowledge to help me make sense of the new application using quotients.
Tuesday, February 8, 2011
Exam Questions, Due February 9
I feel like the most important thing that we have covered so far this semester is rings. There are so many different kinds, and I feel like a lot of the stuff we have learned since learning about rings involves rings, or integral domains, or fields. So knowing the definitions and how to use these things is very fundamental. Maybe that is why I think they are so important; because I don't really understand them that well, or how and when to apply them. Something that I think is important and that I will need to study a lot is the theorems about rings, fields, and integral domains. I need to know these theorems really well so that I will know when they will be useful to me on a problem.
I expect that on the exam, there will be questions involving definitions, maybe fill in the blank or something. I also think that there will be a lot of questions asking for examples of different things we have talked about in class. I think there will also be many proofs. The proofs are what make me the most nervous. I have a hard time understanding where to begin with most proofs, but once I can understand what the question is asking, I can usually figure out how to go about proving it. I also expect that there will probably be lots of questions that are similar to homework questions.
Basically, I think I need to work on understanding the past couple of chapters. I did fine with modular arithmetic and integers, but rings, and isomorphism stuff got really confusing for me. I have had a really hard time understanding what it is talking about and how to apply that to a proof or problem. I need to work on homework from these chapters and learning the theorems to be able to use them on the exam.
I expect that on the exam, there will be questions involving definitions, maybe fill in the blank or something. I also think that there will be a lot of questions asking for examples of different things we have talked about in class. I think there will also be many proofs. The proofs are what make me the most nervous. I have a hard time understanding where to begin with most proofs, but once I can understand what the question is asking, I can usually figure out how to go about proving it. I also expect that there will probably be lots of questions that are similar to homework questions.
Basically, I think I need to work on understanding the past couple of chapters. I did fine with modular arithmetic and integers, but rings, and isomorphism stuff got really confusing for me. I have had a really hard time understanding what it is talking about and how to apply that to a proof or problem. I need to work on homework from these chapters and learning the theorems to be able to use them on the exam.
Sunday, February 6, 2011
4.4, Due on February 7
Difficult
The hardest thing for me in this section was just the idea of the functions that are induced, creating polynomial functions. I can't really tell the difference between polynomials and polynomial functions. Also, how to know if the polynomials are the same function even if they induce the different polynomials. How do you test for this? What number do you have to put into the function to decide if two functions are the same? I am still getting confused by the different meanings of x as well and how to know what x represents in the function or polynomial.
Connections:
This section connects the idea of irreducibility and roots, or factoring. I have been factoring polynomials for a long time, so this part is easier for me. But deciding when a polynomial is irreducible is still difficult for me especially when dealing with the real numbers or modular arithmetic. But, I am fairly comfortable factoring polynomials and finding roots in the integers, so now I just need to think about doing this in different classes of numbers.
The hardest thing for me in this section was just the idea of the functions that are induced, creating polynomial functions. I can't really tell the difference between polynomials and polynomial functions. Also, how to know if the polynomials are the same function even if they induce the different polynomials. How do you test for this? What number do you have to put into the function to decide if two functions are the same? I am still getting confused by the different meanings of x as well and how to know what x represents in the function or polynomial.
Connections:
This section connects the idea of irreducibility and roots, or factoring. I have been factoring polynomials for a long time, so this part is easier for me. But deciding when a polynomial is irreducible is still difficult for me especially when dealing with the real numbers or modular arithmetic. But, I am fairly comfortable factoring polynomials and finding roots in the integers, so now I just need to think about doing this in different classes of numbers.
Thursday, February 3, 2011
4.3, Due on February 4
Difficult:
I had a really difficult time with just the definitions in this section. I don't really understand what an associate is. It sounds to me like it is just a multiple of another thing, but then it seems like all things would be an associate. I just feel like I need to see some examples to really comprehend what this is. The same goes for irreducible. I just can't quite understand what that means. I get that it is like prime, but I feel like if I saw some more examples and had some more exposure to it, then I would understand its meaning better.
Connections:
I found it hard to make connections with the information in this section. I found that theorem 4.11 sounded quite familiar. It seems a lot like some of the theorems for prime numbers. This makes me realize that irreducible is a lot like prime numbers, but I still get confused when applying this concept to polynomials. I guess irreducible polynomials are "prime" polynomials, but that is still confusing for me to relate polynomials to prime numbers.
I had a really difficult time with just the definitions in this section. I don't really understand what an associate is. It sounds to me like it is just a multiple of another thing, but then it seems like all things would be an associate. I just feel like I need to see some examples to really comprehend what this is. The same goes for irreducible. I just can't quite understand what that means. I get that it is like prime, but I feel like if I saw some more examples and had some more exposure to it, then I would understand its meaning better.
Connections:
I found it hard to make connections with the information in this section. I found that theorem 4.11 sounded quite familiar. It seems a lot like some of the theorems for prime numbers. This makes me realize that irreducible is a lot like prime numbers, but I still get confused when applying this concept to polynomials. I guess irreducible polynomials are "prime" polynomials, but that is still confusing for me to relate polynomials to prime numbers.
Tuesday, February 1, 2011
4.2, Due on February 2
Difficult:
I think the hardest thing for me to remember is that the greatest common divisor is monic, or the leading coefficient is 1. I was confused about how you can just multiply by the multiplicative inverse of the leading coefficient to get it to be monic. I wouldn't think of doing that and I feel like that means anything can be a greatest common divisor by multiplying by the inverse of the leading coefficient.
Connections:
I really liked this section. I really like factoring, and I feel like that is what division of polynomials is. I have been factoring polynomials since I can remember in algebra, so it will be good to be able to see how that will apply in abstract algebra as well. I also like finding the GCD, so I am excited to try to find the GCD of polynomials by factoring and finding the highest common term.
I think the hardest thing for me to remember is that the greatest common divisor is monic, or the leading coefficient is 1. I was confused about how you can just multiply by the multiplicative inverse of the leading coefficient to get it to be monic. I wouldn't think of doing that and I feel like that means anything can be a greatest common divisor by multiplying by the inverse of the leading coefficient.
Connections:
I really liked this section. I really like factoring, and I feel like that is what division of polynomials is. I have been factoring polynomials since I can remember in algebra, so it will be good to be able to see how that will apply in abstract algebra as well. I also like finding the GCD, so I am excited to try to find the GCD of polynomials by factoring and finding the highest common term.
Saturday, January 29, 2011
4.1, Due on January 31
Difficult:
I thought one of the most difficult parts of this section was the division algorithm in F[x]. I understood how it works, but I thought the proof was very difficult to follow. I think I got lost in all of the coefficients and exponents. But I thought it was very cool that the division algorithm also works for fields and polynomials. I also got confused about what x was supposed to represent in our polynomials. It talked about how x is not a number that we are trying to solve for, but that it is a specific element of P. I just don't understand what x is.
Connections:
I like that we are working with polynomials now. I understand how to do mathematics on polynomials since I have been dealing with them pretty much since 7th grade. I hope that my background with polynomials will make it easier for me to understand the new applications in this class. I am comfortable adding, multiplying, and dividing polynomials, so I am happy that there is something I am familiar with as I learn some new things that I am just seeing for the first time.
I thought one of the most difficult parts of this section was the division algorithm in F[x]. I understood how it works, but I thought the proof was very difficult to follow. I think I got lost in all of the coefficients and exponents. But I thought it was very cool that the division algorithm also works for fields and polynomials. I also got confused about what x was supposed to represent in our polynomials. It talked about how x is not a number that we are trying to solve for, but that it is a specific element of P. I just don't understand what x is.
Connections:
I like that we are working with polynomials now. I understand how to do mathematics on polynomials since I have been dealing with them pretty much since 7th grade. I hope that my background with polynomials will make it easier for me to understand the new applications in this class. I am comfortable adding, multiplying, and dividing polynomials, so I am happy that there is something I am familiar with as I learn some new things that I am just seeing for the first time.
Thursday, January 27, 2011
Double Bubble Theorem, Presented on January 25
I went to the presentation by Dr. Morgan on Tuesday, and I really enjoyed it. I felt that he was a very entertaining presenter and that I actually understood what he was talking about. The thing that I found most difficult about the presentation was trying to determine which form had the least amount of surface area for the double, triple, quadruple bubbles and so on, even up to 13 or something like that. I don't understand how they would calculate that to find which structure was the best. He had students go up and guess and pick which one they thought had the least surface area, but I just didn't really understand how to even begin to determine that.
I really enjoyed going to the presentation, especially I had heard of the double bubble theorem, but I honestly had no idea what it was. It was great to learn a little bit more about it, especially in such a fun way. Another thing that I really enjoyed was the displays he put on the overhead projector about finding the equilibrium state of the bubbles. He dipped the structure into the soapy water and showed that it found its balance when all three angles forming the connection were the same. (This is really hard for me to explain, but it was cool to see). It was interesting to see the lines of the bubbles move to this equilibrium state just as Dr. Morgan said they would. I learned that I am a terrible guesser since I got most of his questions wrong on the projector, but I also learned a little more about what the double bubble theorem is.
I really enjoyed going to the presentation, especially I had heard of the double bubble theorem, but I honestly had no idea what it was. It was great to learn a little bit more about it, especially in such a fun way. Another thing that I really enjoyed was the displays he put on the overhead projector about finding the equilibrium state of the bubbles. He dipped the structure into the soapy water and showed that it found its balance when all three angles forming the connection were the same. (This is really hard for me to explain, but it was cool to see). It was interesting to see the lines of the bubbles move to this equilibrium state just as Dr. Morgan said they would. I learned that I am a terrible guesser since I got most of his questions wrong on the projector, but I also learned a little more about what the double bubble theorem is.
Response Blog, Due on January 28
On average, the homework assignments take about 2-3 hours. I usually can't seem to answer all of the questions on my own, but by working with others we can talk about what works and what doesn't work and can usually come up with at least a general idea of how to attack the problems. I often feel very confused though. I think I am just not very good at knowing where to start with proofs and I have a hard time understanding what the questions seem to be looking for on my own.
I feel like I understand more from lecture than I do from the readings, but maybe that is because I have done the readings before I go to class, so I have a good idea about what will be covered, so I know what questions I am already trying to understand before lecture starts. I feel that the readings help me prepare for lecture and lecture helps me prepare for the homework. I think so far in this class, lecture and homework has contributed most to my learning. I can focus and learn what is covered in lectures. But I feel that the homework helps me really apply what I have learned to test whether or not I am really understanding.
As far as making my learning more effective, I feel like if I study more as I go I will be able to refer back to previous material and know what theorems, definitions, or other things will help me as I am attempting a proof instead of having to flip through each section of the book to see what will be useful. I think I need to be better about memorizing things as I go instead of cramming it all in before the test. I also learn very well from examples, so I really appreciate all of the examples that we can see in class that will be beneficial for our understanding of the concepts presented.
I feel like I understand more from lecture than I do from the readings, but maybe that is because I have done the readings before I go to class, so I have a good idea about what will be covered, so I know what questions I am already trying to understand before lecture starts. I feel that the readings help me prepare for lecture and lecture helps me prepare for the homework. I think so far in this class, lecture and homework has contributed most to my learning. I can focus and learn what is covered in lectures. But I feel that the homework helps me really apply what I have learned to test whether or not I am really understanding.
As far as making my learning more effective, I feel like if I study more as I go I will be able to refer back to previous material and know what theorems, definitions, or other things will help me as I am attempting a proof instead of having to flip through each section of the book to see what will be useful. I think I need to be better about memorizing things as I go instead of cramming it all in before the test. I also learn very well from examples, so I really appreciate all of the examples that we can see in class that will be beneficial for our understanding of the concepts presented.
Tuesday, January 25, 2011
3.3, Due on January 26
Difficult:
I thought the beginning of the chapter was very difficult. I got confused with all the relabeling and and new notation of the bars. I also have a hard time remembering what injective and surjective means and how to prove that a function is injective and surjective. I think the most difficult part of this section for me will be to use the notation and to show injectivity and surjectivity of functions.
Connections:
I remember proving that something is injective, surjective, and bijective in my 290 and 341 classes, but I never really understood what it means, especially how to prove it, and to know if you have actually proved it. I have seen the function notation and read the definitions of these things many times, but I always have a hard time remembering them. So hopefully I will be able to remember what I had been taught and be able to create proofs using some of my previous knowledge of injectivity and surjectivity.
I thought the beginning of the chapter was very difficult. I got confused with all the relabeling and and new notation of the bars. I also have a hard time remembering what injective and surjective means and how to prove that a function is injective and surjective. I think the most difficult part of this section for me will be to use the notation and to show injectivity and surjectivity of functions.
Connections:
I remember proving that something is injective, surjective, and bijective in my 290 and 341 classes, but I never really understood what it means, especially how to prove it, and to know if you have actually proved it. I have seen the function notation and read the definitions of these things many times, but I always have a hard time remembering them. So hopefully I will be able to remember what I had been taught and be able to create proofs using some of my previous knowledge of injectivity and surjectivity.
Saturday, January 22, 2011
3.2, Due on January 24
Difficult:
I had a hard time understanding the definition of a unit. I got confused because I sounded to me like every ring with identity may not always have a unity, but I thought that was the definition of being a ring with identity. If a ring has identity, then it has a multiplicative inverse. But the definition made it sound like a ring with identity only has a unit if there exists and element a such that ua=1=au. I just was confused if there is a unit for every ring with identity.
Connection:
I really liked this section. I like that subtraction in a ring is defined as most people think of subtraction (just adding a negative). I like that arithmetic in rings, for the most part, is something that I am familiar with. For example, theorem 3.5 made a lot of sense to me. I like when I am able to follow the logic of the proofs and understand the meaning of the theorems.
I had a hard time understanding the definition of a unit. I got confused because I sounded to me like every ring with identity may not always have a unity, but I thought that was the definition of being a ring with identity. If a ring has identity, then it has a multiplicative inverse. But the definition made it sound like a ring with identity only has a unit if there exists and element a such that ua=1=au. I just was confused if there is a unit for every ring with identity.
Connection:
I really liked this section. I like that subtraction in a ring is defined as most people think of subtraction (just adding a negative). I like that arithmetic in rings, for the most part, is something that I am familiar with. For example, theorem 3.5 made a lot of sense to me. I like when I am able to follow the logic of the proofs and understand the meaning of the theorems.
Thursday, January 20, 2011
3.1, Due on January 21
Difficult:
In this section of 3.1, I had a hard time understanding why you would have to have 4 new axioms to determine whether or not the subset of a ring would be a subset. If its a subset of a ring, I don't understand why these 4 axioms need to be re-fulfilled. And if these 4 do, why don't the other 4? I don't understand why closed under addition and multiplication, and a 0 element is in the subset, and that there is a solution to a + x = 0 ( the 0 element in the subset). I didn't understand why these are the 4 axioms that need to be fulfilled.
Reflection:
I really like using the Cartesian product of z modulo 6 and z modulo 4 to see if it is a ring. I like being able to write out addition and multiplication tables to organize the information to help determine if the Cartesian product is a ring. I also like that we can redefine addition and multiplication to find other rings and use the axioms to see if the set is a ring or if that ring has any subrings.
In this section of 3.1, I had a hard time understanding why you would have to have 4 new axioms to determine whether or not the subset of a ring would be a subset. If its a subset of a ring, I don't understand why these 4 axioms need to be re-fulfilled. And if these 4 do, why don't the other 4? I don't understand why closed under addition and multiplication, and a 0 element is in the subset, and that there is a solution to a + x = 0 ( the 0 element in the subset). I didn't understand why these are the 4 axioms that need to be fulfilled.
Reflection:
I really like using the Cartesian product of z modulo 6 and z modulo 4 to see if it is a ring. I like being able to write out addition and multiplication tables to organize the information to help determine if the Cartesian product is a ring. I also like that we can redefine addition and multiplication to find other rings and use the axioms to see if the set is a ring or if that ring has any subrings.
Tuesday, January 18, 2011
3.1, Due on January 19
Difficult:
I thought the most difficult part of this section was keeping track of all the different types of rings. I had a hard time distinguishing when a ring is commutative, is a ring with identity. And then once I figure out if its commutative, then it could be an integral domain or a field. I didn't really understand these definitions and how best to keep track of them. I also didn't really understand what the 0 with a subscript R meant, or a 1 with subscript R. I think in this section I mostly got lost in all the definitions and new notation.
Connection:
As with modular arithmetic, I like that rings follow properties of addition and multiplication like the integers. It makes it easier for me when I already understand the axioms that define a ring. So once I can better understand the new notation, I will feel more comfortable with recognizing rings because I already know about the axioms, since I have been using these axioms pretty much all of my life.
I thought the most difficult part of this section was keeping track of all the different types of rings. I had a hard time distinguishing when a ring is commutative, is a ring with identity. And then once I figure out if its commutative, then it could be an integral domain or a field. I didn't really understand these definitions and how best to keep track of them. I also didn't really understand what the 0 with a subscript R meant, or a 1 with subscript R. I think in this section I mostly got lost in all the definitions and new notation.
Connection:
As with modular arithmetic, I like that rings follow properties of addition and multiplication like the integers. It makes it easier for me when I already understand the axioms that define a ring. So once I can better understand the new notation, I will feel more comfortable with recognizing rings because I already know about the axioms, since I have been using these axioms pretty much all of my life.
Thursday, January 13, 2011
2.3, Due on January 14
Difficult:
For the most part, I felt like I pretty much understood this section, which is a pretty big surprise for me! However, the part that I found to be most difficult was in the proof of Corollary 2.9. In the last part of the proof when the authors are proving uniqueness, they have this great idea to multiply both ends of the equation by u. This made sense, but the reason I found it to be so difficult was because I would never be able to think of doing something like that. I see why it works, but when I am doing proofs, I never think of doing something like this. Multiplying through by the number I want to both sides of the equation is not something that I ever think to do. This step in the proof was difficult for me to understand only because it is not something that I ever think of doing.
Connection:
I really liked the method used to solve the equation ax=b in Z mod n using the Euclidean algorithm to find a linear combination. Then using the solutions in the linear combination, you can find the solution to ax=b. I like being able to use algebra skills that I am already familiar with to solve new mathematics that I encounter in this course. Even though modular algebra is a newer, more difficult concept to me, I feel like I better understand the material when I can apply concrete ideas that I already know how to use.
For the most part, I felt like I pretty much understood this section, which is a pretty big surprise for me! However, the part that I found to be most difficult was in the proof of Corollary 2.9. In the last part of the proof when the authors are proving uniqueness, they have this great idea to multiply both ends of the equation by u. This made sense, but the reason I found it to be so difficult was because I would never be able to think of doing something like that. I see why it works, but when I am doing proofs, I never think of doing something like this. Multiplying through by the number I want to both sides of the equation is not something that I ever think to do. This step in the proof was difficult for me to understand only because it is not something that I ever think of doing.
Connection:
I really liked the method used to solve the equation ax=b in Z mod n using the Euclidean algorithm to find a linear combination. Then using the solutions in the linear combination, you can find the solution to ax=b. I like being able to use algebra skills that I am already familiar with to solve new mathematics that I encounter in this course. Even though modular algebra is a newer, more difficult concept to me, I feel like I better understand the material when I can apply concrete ideas that I already know how to use.
Tuesday, January 11, 2011
2.2, Due on January 12
Difficult:
At the beginning of the section, it defined classes of integers, A, B, C, D, E, where the classes were all disjoint, and every integer was in one of the equivalence classes, but it showed that B+C=D and B+C=A. I was confused why this happened. Did they set up these equivalence classes in such a way just to prove this point? I was just really confused where these equivalence classes came from, and why the addition property didn't work in them. I had a difficult time understanding when this would happen in a problem, and why the addition and multiplication that were defined at the beginning of the chapter were just "tentative".
Connection:
The most interesting part of this section for me was that modular arithmetic also follows the same properties as arithmetic in the integers (with the exception that if ab=0, then a=0 or b=0). I thought that this section was really cool to see that modular arithmetic will follow the same rules we have been taught over and over in math. These rules are ones that we have practiced over and over, so it is interesting to be able to apply them to a new kind of arithmetic. I really like when I can use something that is familiar to me when I am trying to learn something new and more difficult.
At the beginning of the section, it defined classes of integers, A, B, C, D, E, where the classes were all disjoint, and every integer was in one of the equivalence classes, but it showed that B+C=D and B+C=A. I was confused why this happened. Did they set up these equivalence classes in such a way just to prove this point? I was just really confused where these equivalence classes came from, and why the addition property didn't work in them. I had a difficult time understanding when this would happen in a problem, and why the addition and multiplication that were defined at the beginning of the chapter were just "tentative".
Connection:
The most interesting part of this section for me was that modular arithmetic also follows the same properties as arithmetic in the integers (with the exception that if ab=0, then a=0 or b=0). I thought that this section was really cool to see that modular arithmetic will follow the same rules we have been taught over and over in math. These rules are ones that we have practiced over and over, so it is interesting to be able to apply them to a new kind of arithmetic. I really like when I can use something that is familiar to me when I am trying to learn something new and more difficult.
Saturday, January 8, 2011
2.1, Due on January 10
Difficult:
I understood the congruence classes, but I got confused about the second part of Corollary 2.5. It says that there are exactly n distinct congruence classes. I am confused about the meaning of distinct. Does this just mean they do not overlap with each other? I also found it difficult to understand that the set of all congruence classes has exactly n elements. I was confused because at first it points out that the congruence class of 2, 5, -1, and 14 are the same modulo 3, but that only the congruence class of 2 is in the set of all congruence classes modulo 3. I just didn't understand why all of the other ones wouldn't be considered to be in the set of congruence classes modulo 3 because they are the same as the congruence class of 2 modulo 3. I guess overall, the most difficult part of this section for me was understanding congruence classes and properties of them.
Reflection:
I really like that the reflexive, symmetric, and transitive properties also apply to congruence modulo n. I know that as a secondary mathematics teacher, I will encounter these three properties quite often when teaching algebra and other subjects. I like that these are properties that I am familiar with and understand so that I can apply the information that I already understand, and will one day teach, to something that I am less familiar with, congruence and congruence classes. I am excited to be able to teach my students about these properties and how they apply in algebra classes.
I understood the congruence classes, but I got confused about the second part of Corollary 2.5. It says that there are exactly n distinct congruence classes. I am confused about the meaning of distinct. Does this just mean they do not overlap with each other? I also found it difficult to understand that the set of all congruence classes has exactly n elements. I was confused because at first it points out that the congruence class of 2, 5, -1, and 14 are the same modulo 3, but that only the congruence class of 2 is in the set of all congruence classes modulo 3. I just didn't understand why all of the other ones wouldn't be considered to be in the set of congruence classes modulo 3 because they are the same as the congruence class of 2 modulo 3. I guess overall, the most difficult part of this section for me was understanding congruence classes and properties of them.
Reflection:
I really like that the reflexive, symmetric, and transitive properties also apply to congruence modulo n. I know that as a secondary mathematics teacher, I will encounter these three properties quite often when teaching algebra and other subjects. I like that these are properties that I am familiar with and understand so that I can apply the information that I already understand, and will one day teach, to something that I am less familiar with, congruence and congruence classes. I am excited to be able to teach my students about these properties and how they apply in algebra classes.
Thursday, January 6, 2011
1.1-1.3, Due on January 7
Difficult:
For me, the most difficult part of this section was in section 1.3. In the fundamental theorem of arithmetic, it states how every integer is unique. I was very confused by how p could be a plus or minus q. I just couldn't understand what this statement was saying, and how it made each prime factorization unique. I think I just got lost in all the plus or minus signs. I had a really hard time trying to keep track of them, and whether or not they were canceling out, or if the answers were negative or positive. I found the plus and minus signs in the Fundamental Theorem of Arithmetic and its proof to be very confusing and the most difficult part of this section of reading.
Reflection:
I really liked section 1.3. It reminded me of what prime numbers are and that all numbers can be factored into a product of primes. This is something that as a secondary math teacher, I will encounter a lot as I teach high school and middle school students. As I gain a better understanding of prime and prime factorization, I will be able to better articulate these ideas to my future students. Also, in section 1.2, I liked learning about relatively prime. I made a task in one or my previous education classes that used the concept of relatively prime to solve the problem, so it was nice to see this concept come up again in this class. I plan to use my task in my future algebra classes, so I will also teach my students about what it means for two numbers to be relatively prime which was discussed in this abstract algebra book.
For me, the most difficult part of this section was in section 1.3. In the fundamental theorem of arithmetic, it states how every integer is unique. I was very confused by how p could be a plus or minus q. I just couldn't understand what this statement was saying, and how it made each prime factorization unique. I think I just got lost in all the plus or minus signs. I had a really hard time trying to keep track of them, and whether or not they were canceling out, or if the answers were negative or positive. I found the plus and minus signs in the Fundamental Theorem of Arithmetic and its proof to be very confusing and the most difficult part of this section of reading.
Reflection:
I really liked section 1.3. It reminded me of what prime numbers are and that all numbers can be factored into a product of primes. This is something that as a secondary math teacher, I will encounter a lot as I teach high school and middle school students. As I gain a better understanding of prime and prime factorization, I will be able to better articulate these ideas to my future students. Also, in section 1.2, I liked learning about relatively prime. I made a task in one or my previous education classes that used the concept of relatively prime to solve the problem, so it was nice to see this concept come up again in this class. I plan to use my task in my future algebra classes, so I will also teach my students about what it means for two numbers to be relatively prime which was discussed in this abstract algebra book.
Wednesday, January 5, 2011
Introduction, Due on January 7
My name is Haley Stevenson. This is my 4th year here at BYU, so I am technically a senior. However, I still have one year left in school before I will graduate. Beyond calculus, I have taken Linear Algebra, Multivariable Calculus, Theory of Analysis, Math 290, and am currently enrolled in Differential Equations. I am taking this Abstract Algebra class not only because it is a requirement for my math education major, but I really like algebra so I hope this course will help me better understand the kinds of algebra that I will teach in my future classroom.
One math professor that I had that was a very effective teacher truly cared about the success of his students. I really appreciated that he took extra time when necessary to explain difficult concepts. I also appreciated that he would show examples in class of what to expect on homework and exams. I find it difficult in courses when I feel that the homework or exams are completely different than what is covered in a class. I also liked that he was willing to take extra time to help students through office hours and such to further explain and help solidify concepts or go over difficult homework problems.
The only thing that I find to be unique about myself is that I am semi-left handed. I write with my left hand, but I do all other things right handed (eating, painting my nails, playing sports, brushing my teeth, etc). I'm not sure why this is... but that is just how I am I guess!
One math professor that I had that was a very effective teacher truly cared about the success of his students. I really appreciated that he took extra time when necessary to explain difficult concepts. I also appreciated that he would show examples in class of what to expect on homework and exams. I find it difficult in courses when I feel that the homework or exams are completely different than what is covered in a class. I also liked that he was willing to take extra time to help students through office hours and such to further explain and help solidify concepts or go over difficult homework problems.
The only thing that I find to be unique about myself is that I am semi-left handed. I write with my left hand, but I do all other things right handed (eating, painting my nails, playing sports, brushing my teeth, etc). I'm not sure why this is... but that is just how I am I guess!
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