Section 8.3 Due June 14th

1. Wow there are so many proofs to remember. I hope that I can use then sufficiently on the test. I feel like the middle sections of chapter 7 and this section are the very hardest out of everything. I liked the cycles the best.
2. The properties of the order of groups are so interesting. Who is this Sylow person?

Sections 8.1-8.2 Due June 8th

1. I didn't understand how on the bottom of page 245 proving that the elements of Z_6 can be written as a sum of elements of M and N in one and only way proved that they were normal subgroups. The proofs in section 8.2 were very long and complicated, so I wonder whether I will be able to use them sufficiently in homework problems. I'm not understanding how we go about finding the elementary and invariant divisors.
2. It was interesting how prime power orders have such an effect on the groups and subgroups.

Sections 7.9-7.10 Due May 6

1. I don't understand how the product of the cycle groups works.I didn't understand the proof of lemma 7.54, especially when all the sudden they had 12k. I didn't see where that came from.
2. I really liked section 7.9 and thought it was kind of fun, but it seems a little pointless and like something that someone just made up and gave a name. I see no relevance in all of the connections that were made between cycles and the group S_n. They were interesting theorems, but seemed useless.

Sections 7.6-7.8 Due June 4

1. Wow the whole section on quotient groups and homomorphisms was confusing. I feel like we just keep building and building upon the original group idea and it's getting so complex that if I'm confused about a little thing in one section, then it's really detrimental in the next. Especially since we're moving so fast.
2. It's interesting that every subgroup of an abelian group is normal. I really liked quotient rings, but quotient groups are a lot more abstract.
   

Section 7.5 Due June 2

1. When it changes from {b in G such that ba^-1 in K}, to {b in G such that b = ka} it makes it hard to grasp what the set actually is talking about. The new notation H + 2 for <3> doesn't make sense. Theorem 7.25 was hard to follow.
2. I thought that Thm. 7.29 about how every group of order 4 is isomorphic to Z_4 or Z_2 X Z_2 was interesting. I don't like how instead of giving full proofs they say just to copy the proofs in previous chapters "with obvious notation changes".

Section 7.3-7.4 Due May 28

1. I'm having a hard time understanding cyclics. And I can't figure out what an automorphism would be with groups like S_3 and D_4. It started getting hazy when it was talking about the image and e_G and e_H, so if we could go over that a lot in class that'd be great.
2. It's interesting that every single infinite cyclic group or order n is isomorphic to Z_n. It was interesting how similar the properties of proofs for subgroups were to subrings.

Sections 7.1-7.2 Due May 26

1. Does part 2 of theorem 7.5 imply that G is commutative, and hence an abelian group? When it started talking about order of an element, I didn't understand what e was. When it says "if a has a finite order n" is n the number of times you have to multiply a to get e? and therefore is n not unique?
2. Groups seem to follow traditional rules with exponentials, and overall still seem fairly simple. It's interesting that we didn't talk about order in rings, but only in groups. The corollary on the top of page 178 was interesting because I wouldn't have thought that such a correlation between the orders of elements would exist.

Section 9.4 & 7.1 Due May 24

1. The definition of addition where [a,b] + [c,d] = [ad+bc, bd] was very random. Theorem 9.31 made no sense. So with groups can you just define * as whatever you want it to be? Because composition doesn't seem like just a standard operation like multiplication of addition or something. I don't get what the actual group is. Is it the set of the bijective compositions? Or the set of the permutations?
2. It was interesting how on the proof of lemma 9.29 they proved it backwards and went from f(a)+f(b) to f(a+b) instead of the other way around. I haven't seen it like that before. Groups seem similar to rings, but with less operations.

Sections 6.2-6.3 Due May 20

1. The definition of prime ideals was a little strange. I didn't understand how there could be infinitely many maximal ideals in Z, nor why the maximums have to be prime. I couldn't remember what a kernal was so all of the maps to the kernal K didn't make sense.
2. I like the idea of ideals and how the new notation is easier. I still have a hard time seeing applications of any of this though. It said something in the book about the Chinese Remainder Theorem, which sounds like something interesting.

Section 6.1 Due May 19

1. I wasn't really sure what they meant by "absorbs". Theorem 6.6 was a little confusing. Are there infinitely many ideals? Finitely generated was also confusing.
2. So according to theorem 6.1, a subset can be an ideal without necessarily being a subring? Ideals seem to make modular arithmetic a little simpler. It'll be interesting to see how these will add to what we have been doing, since they seem to just be a replacement of our current notation of classes.

Sections 5.1-5.3 Due May 14

1.  The extensions didn't make any sense. Especially after they replaced x^2 with x+1. The whole extension field idea was confusing.
2. Everything else seemed exactly like normal equivalence classes. The proofs were the same. The multiplication and addition tables worked the same. I think the idea of equivalence classes with something like polynomials is a little bit silly though. It seemed like it was just taking a simple enough idea and making it as complicated as possible by applying it to polynomials rather than just integers. However I would be interested to here how something like this would be useful.

Sections 4.3-4.4 Due May 14

1. It will be hard to get to this new way of thinking about roots. Like when it says that a root is an induced function from R to R that maps a to O. Also I didn't understand what corollary 4.19 on the bottom of page 4.4 was saying.
2. The definition of primality or irreducibility was just like with normal integers. As was the unique factorization. The proofs were similar too. I think that this section seemed pretty straight forward and similar to what we've already done.

Sections 4.1-4.2 and App. G Due May 11

1. Long division with polynomials is very hard for me. Also the proof of the division algorithm for polynomials was a little overwhelming. Over all it seems like it will be hard keeping track of everything in these proofs because they are so big and there as so many variables.
2. I noticed that these sections were almost identical to the sections of normal integers. The division algorithm, the euclidean algorithm, definitions of division and gcd's, everything was similar to the normal steps we take with integers.

Section 3.3 Due May 9

1. Everything was familiar until the bottom of page 72 when it started talking about a homomorphism of rings. Then I was totally lost. I didn't understand what theorem 3.12 was or the proof of it was talking about. Also it seems like it will be pretty hard to find an isomorphism or homomorphism between to rings.
2. I learned most of this in my linear algebra class, but with vector spaces, not rings. As of right now I'm having a hard time seeing a real world application for this section, interesting as it is. I guess we do create bijections all the time when we pair things up and correlate things.

Section 3.1-3.2 Due May 6

1. For some reason I'm still confused about what an integral domain is, and how if it is finite, it is a field. In theorem 3.6 it says that a subset of a ring is a subring if it is closed under subtraction and multiplication, but before it was addition and multiplication. Does it have to be all three or just either subtraction or addition?
2. The properties seem pretty usual. The exponent rules are the same as for the reals, as well as squaring a+b. None of the properties of rings seem out of the ordinary. The fact that there are many units in the matrix ring M(Z) is interesting.

Section 2.3-3.1 Due May 4

1. I didn't understand what the integral domain ring was. I was also confused by the definition of a field, because I'm used to my linear algebra teacher's definition. But maybe these two aren't the same thing. Also the example on page 43 of the set T= (r,s,t,z) was so much different then the other examples, and the addition and multiplication in the tables was defined strangely.
2. It's interesting how much these rings remind me of vector spaces. The definition is almost the same if not the same, as well as the properties. Also, a subring seems just like a subspace. Are these simply two names for the same concept?

Sections 2.1-2.2 Due May 2

1. Congruence classes have always been hard for me. Especially the arithmetic. The multiplication and addition charts are easy enough, but for some reason I have a hard time using the properties of equivalence classes in proofs. It gets so hard to wrap my mind around all the a's, b's, c's and n's. It's difficult to see which properties to use to get to what we want at the end of a proof.
2.  Well this definitely relates to changing the base of a number, and also, like I said in my last post, to sending messages through computers and to binary code. It's interesting that the associative, commutative, additive, etc. that we learned in middle school apply to not just normal arithmetic, but also to modular arithmetic.

Section 1.3 and Appendix D Due April 29

1. It was difficult for me to understand Appendix D when it was talking about Rings and Groups because we haven't learned that yet. I felt very confident in Math 290 proving equivalence relations, but I was never really good at understanding equivalence classes, and understanding how to show that two are equal, or anything like that. The proof about the uniqueness of a prime factorization in 1.3 was also a little hard to follow. 
2. I would be very interested to see how these things apply to the real world. I know that there are interesting things that have to do with prime numbers in the Fibonacci sequence. Modular arithmetic I know is used in computer programing, like with Z_2 binary code. It's very cool how we can send and receive messages with the use of very large prime numbers and public and private keys.

Sections 1.1-1.2 Due April 28

1. The most difficult part of reading this section was following the proofs. I understand that they arrive at the correct result, but it's difficult for me to see where the ideas are coming from, and how one would know where to start and what equations/symbols/substitution etc. one needs to use in order to eventually arrive at a correct result and a complete proof. I suppose it comes with practice.
2. I actually am familiar with all of the material in these sections thanks to my Math 290 class. One of my favorite things in that class was the Euclidean Algorithm. I think it is the coolest thing because I've always had to write out the divisors of both, and find the greatest one they have in common. I wish they would teach this in high school or even jr. high because it makes finding the gcd so much simpler. It also is very similar to when we are trying to change the base of a number, which can help if we are using some other form of number system. That's something we learned in my history of math class.

Introduction Due April 28

• I am a sophomore majoring in Mathematics Education.
• I have taken Math 290, 300, and 313. 
• I am taking this class because it is required for my major. 
• Dr. McKay here at BYU was my Calculus II professor and he was the most effective because he was great at explaining things. He not only taught us the computational procedures, but where they can from and why they work. Even with doing that he still made everything seem simple. Dr. Chahal for Math 313 was the least effective because his lectures had no relation to the homework he assigned, so our classmates had to get together often out of class and just try to figure out what the questions were asking. Also, the book he wrote for us to use was very confusing.
• I was Taylorsville's Jr. Miss 2009, which is like a scholarship program/pageant type thing that I won for my city. 
• I can come to the Thursday office hour, and maybe the MW afternoon as well.