Final Reflection, due on 4/13

• Which topics and theorems do you think are important out of those we have studied?
• It depends on your definition of important.  I don't foresee the topics covered in this course being very useful in my research.  It may some day become relevant to my research but we'll see.
• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Wednesday.
• I just have to review everything.  Nothing was too difficult.  It was just a lot of material.  So I have to make sure I remember the little computation problems and how to apply the big theorems.
• One thing I have to go over is ideals, maybe principal ideals in particular.
• How do you think the things you learned in this course might be useful to you in the future
• I'm not sure.  See first answer.

8.3, due on 4/11

1.  No love for proofs of Sylow's Theorems, sad day.  Corollary 8.18 seemed very obscure but the proof was straightforward.

2.  The First Sylow Thm was interesting but they were too lazy to prove it.  Cauchy's Theorem was the same thing but k=1.

8.2, due on 4/8

1.  The notation review was fine but I still am not sure what the big deal is about the difference between addition and multiplication.  The proof of Lemma 8.6 was pretty beastly.

2.  Theorem 8.5 was pretty cool and the proof wasn't too bad.  The Fundamental Theorem of Finite Abelian Groups seems pretty fundamental.

8.1, due on 4/6

1.  I'm not sure why this is new chapter it seems like the same stuff we've been doing.

2.  The order of the Cartesian product of groups was very simple.  Lemma 8.2 was pretty nifty.  The proof was very straightforward.

7.10, due on 4/4

1.  At first I thought it was the simplicity of An conjugate transpose but then I noticed that it was a footnote.  I forgot what simple meant.  The proofs in this section were a little hard to follow.

2.  The lack of examples was interesting.  I was surprised this author loves examples.

7.9, due on 4/1

1.  I did not follow the proof of Corollary 7.47 very well.

2.  I really liked the new k-cycle notation because I hated writing out the long things every time in Sn.  The concept of disjoint was very straightforward.  This stuff seems to getting closer to graph theory/network stuff.  Or I'm just barely beginning to see the connection.

7.8, due on 3/30

1.  The idea of the kernel was weird to me because I think of a kernel as the set that is mapped to zero.  But since groups don't have a zero element I guess the next closest thing is the identity.  They skipped the Second Isomorphism Theorem for Groups!

2.  All of the theorems were very straightforward.  I was perfectly okay with the proofs.

7.7, due on 3/28

1.  Theorem 7.37 was a little weird.

2.  It seems like quotient groups are an extension of quotient rings.  Theorem 7.35 was very straightforward and the proof was easy to follow.

7.5, due on 3/25/11

1.  I don't understand why no one other than Jimmer scored in double figures on our team.

2.  Theorem 7.33 was very straightforward.  Theorem 7.34 made lots of sense.  Dr. Doud already proved it in class on Wednesday.

7.6, due on 3/22

1.  I don't understand why they introduce right cosets then left cosets a couple sections later.

2.  The idea of left congruence was pretty simple.  And the idea that it was an equivalence relation was very simple.  The idea of a normal group was very straightforward.

Midterm 2 Response

• Which topics and theorems do you think are the most important out of those we have studied?
• Langrange Theorem
• The general ideas of group theory
• Quotient rings
• What kinds of questions do you expect to see on the exam?
• Well I can tell you one thing I hope we don't have to play sudoku in the testing center.
• A proof or two, even though I like the proofs more
• Judging from the last test we'll probably have to give some examples of different things like finite and infinite groups, subgroups, cosets, etc.
• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Monday.
• I just need to review the concepts more, go over the computational/homework type problems, and look over the important theorems some more.

Math in the Movies

This talk was ok but it wasn't that in depth.  He lightly scratched the surface of some topics that sounded interesting but it was more of a general overview of the movie making process which was boring to me.  He was the man guy who made the chess movie at the beginning of toy story I believe.  That was pretty awesome.  He also had clips of other movies that we're pretty good.

7.5, due on 3/18

1.  The proofs were pretty easy but I didn't realize before the difference between Z4 and Z2xZ2  and between Z6 and S3 but now it's very clear.

2.  This section was pretty straightforward.  The theorems seem very useful.  If you know the order of a group you can tell a lot about it if the order is prime, 4, or 6.

7.5, due on 3/16

1.  The only part of the proof of theorem 7.25 that wasn't clear was that "b in G imples b = eb in Kb is a subset of the union of right cosets implies G is a subset of the union of the right cosets."  The red part was unclear.  Is it because b was an arbitrary element of G?  In theorem 7.26 does [G:K] mean the number of distinct cosets?  They explain that in the proof but it seems like they should have explained it before they used it.  If they did I missed it.

2.  The first three results were very straightforward and easy to understand.  Theorem 7.25 was very interesting and the proof was pretty clear.  Apart from the [G:K] part theorem 7.26 was very clear.  I liked the second part of corollary 7.27.

7.4, due on 3/14

1.  Theorem 7.19 was strange to me.

2.  I feel like the 3rd or 4th section of every chapter of this book is called isomorphisms and homomorphisms.  The ideas were very straightforward and sort of repetitive.  Theorem 7.18 was very straightforward and the proof understandable.

Curing Cancer with Math Lecture

First of all, sorry I took so long to write this.  I understand if you don't except it.

Second, I really enjoyed this talk.  I work in IDeA Labs in the CS Dept and we do a lot of mathematical modeling.  One of my friends studied predator-prey models and developed one for leaf cutter ants.  So when the speaker started talking about applying them to cancer I was very interested and followed her quite easily.  It was pretty amazing how she was able to develop the model then optimize the medicine dosage in order to help patients.

7.3, due on 3/11

1.  Theorem 7.11 was a little confusing.  I guess I didn't know that the set being finite implied all the elements were of finite order.

2.  The idea of a subgroup was very straightforward.  The examples were simple enough.  I liked the matrix example on pg 182.  The idea of cyclic subgroups was interesting.

7.4, due on 3/7

1.  Most of the ideas made sense.

2.  When I read the definition of a group I thought it was pretty close to some of the axioms of a ring so Theorem 7.1 was straightforward.  Corollary 7.3 was a clear corollary and it made sense.  Theorem 7.4 made sense and the proof probably isn't too rough.

7.3, due on 3/4

1. I was thrown off that it was not built upon an old idea.  It was sort of unique.

2.  The definition of group was pretty straightforward.  And it was kind of related to the old ideas, that is the axioms for a group are similar to those of a group.

6.3, due on 3/1

1. The idea of a maximal seemed strange to me and I'm not sure of its utility.  The proof of theorem 6.15 was a little much.

2.  Expanding Prime to the structure of R/I is interesting.  Theorem 6.14 was pretty straightforward.  Cor. 6.16 was interesting and the proof was practically trivial.

6.2, due on 2/28

1.  The concept of the kernel is very familiar but can it also be called the nullspace?  Theorem 6.13 was pretty heavy but I think the post-theorem paragraph made it clear.

2.  Theorem 6.12 was very logical and the proof was a little more complicated/long than I thought it would have to be.  The proof of 6.13 made sense but did not give much insight into the importance of the theorem.

6.2, due on 2/25

1.  The book is very good at building on stuff it has already done but I feel like it falls short in explaining how these concepts are more abstract.  I think it makes sense but it is not exactly clear to me why the quotient ring is any different.  It feels we gave a new name and notation to something that already had a name and coefficient.

2.  The explanation of theorem 6.8 and definition of addition and multiplication of cosets was very straightforward.  Theorem 6.9 seemed trivial and the definition of quotient ring seemed redundant.

6.1, due on 2/23

1.  The definition of coset was unclear to me but the examples helped somewhat but not completely.

2. Theorem 6.4 seemed very logical and the proof very straightforward.  The same thing with theorem 6.5.

6.1, due on 2/22

1. The ideas in abstract algebra are getting more abstract, go figure.  I feel like it has been developed well so it is not that hard to follow.  I did not follow the example after principal ideal generated by c part.

2.  The idea of an ideal is interesting.  The examples depicted the point very well.  Theorem 6.1 was very straightforward and the proof was practically trivial.  Theorem 6.2 also seems pretty trivial/obvious.

5.3, due on 2/18

1.
All the proof were straightforward and/or cop-outs from previous chapters.

2.
In the first example they kind of used the trick we learned in class but they didn't explain it very well, tsk tsk.  It is interesting that F[x]/(p(x)) contains a root of p(x). The discussion of how R[x]/(x^2+1) is isomorphic to (or a definition of ) the complex numbers was very interesting.

5.2, due on 2/16

1. Theorem 5.7 seems strange to me.  I followed the proof I guess I just don't understand the importance of it.

2. Theorem 5.8 was a cop out.  It is the same thing as 5.7.  Theorem 5.9 was pretty interesting.  This section was pretty weak.  It was short and not very interesting.  At least it was easy to follow.

5.1, due on 2/14

1. This section seemed almost completely trivial to me.  Cor 5.5 took a couple seconds to decipher what it was trying to say but I got it.

2.  So we're combining chapter 2 with chapter 4.  This made all the proofs pretty much trivial.  Cor. 5.5 was slightly interesting.  The set of congruence classes of modulo p(x), degree n, is a subset of S, the set of polynomials degree less than n.  I think they worded it strangely.  I like my explanation better.

9.4, due on 2/11

1.  I thought it was strange how we jumped five chapters and only for one chapter.  I'm not sure why they call it the field of quotients when they could just call it the field of rationals but whatever.  I'm not sure that I fully understand Theorem 9.31 or its importance.

2.  I feel like Theorem 9.31 is very important but I'm not quite getting it.  I also feel like we went from being simple to abstract back to simple again.

Test Reflection, due on 2/9

• Which topics and theorems do you think are the most important out of those we have studied?
• The ones with names are almost always the most important, so the well-ordering axiom division algorithm, the Euclidean algorithm, the prime number theorem, the remainder theorem, and the factor theorem.
• What kinds of questions do you expect to see on the exam?
• I plan to see the questions like the homework with at least one proof of a theorem from the book, such as the remainder and factor theorems.
• What do you need to work on understanding better before the exam?
• I just need to review it all, especially a couple of the tricky computational problems from towards the beginning of the semester.

4.3, due on 2/4

1.  I'm not sure if I understood the definition of a unit before.  It seemed trivial to me but Theorem 4.8 seemed very clear and non-trivial to me.

2. I thought it was interesting that in fields irreducible is prime and reducible is composite.  I also thought it was interesting that every polynomial of degree 1 is irreducible, which means that in a field there are an infinite number of primes (irreducibles).

4.2, due on 2/2

1. It wasn't completely clear whether or not monic meant only the leading coefficient is 1 on just the highest degree polynomial or all of the non constant ones.  Also is 5 monic or is 1 monic?

2.  I thought it was interesting how most of the proofs were one-liners because they were the same as chapter 1 proofs.  I also thought it was interesting how again the simplistic, practically trivial stuff from chapter 1 applies directly to these abstract fields.

4.1, due on 1/31

1.  I found the xa=ax for every a in R part hard to believe.  They don't give a proof of it in the section and they don't give it in the appendix they refer to either, no good.

2.  I thought it was interesting that now we're going to expand to polynomials in rings/fields.  I found the division algorithm applied to fields interesting.  The proofs (that were given) were understandable.

Class Feedback, due on 1/28

• How long have you spent on the homework assignments?  I spend about an hour on the homework assignments.
• Did lecture and the reading prepare you for them? Yes.
• What has contributed most to your learning in this class thus far?  The homework helps.
• What do you think would help you learn more effectively or make the class better for you? Since the reading is mandatory and mostly we just go over the reading in class lately I've been getting bored in class.  So, maybe less reading or different stuff in lecture than in the reading, but I understand that for some people seeing it twice helps them understands.  So I'm not sure.

3.2, due on 1/26

1.  It is a little unclear to me how to show how rings are not isomorphic because how do you show there is no possible function.

2.  I like the idea that now we have functions between our rings.  This will make things more interesting.  It reminds me of switching between the frequency domain.  Pretty much all of the examples were straightforward.  The proofs also were fairly easy to follow.

3.2, due on 1/24

1.  I was a little confused by Theorem 3.5 (7) because on Friday's homework we found a 1r=0 so how would -1ra=-a?  Oh, well by looking at that proof I got it.

2.  I liked the idea that these principles could be applied to any ring.  It seems powerful.  Also all the proofs were very straightforward.  I liked Theorem 3.6 for checking if a subset was a subring because Theorem 3.2 seemed like too much stuff to remember.

p.s. The homework assignment due Friday was a little excessive.  I do not think it was necessary to do addition and multiplication tables for three separate Cartesian products.  After two, maybe even one, it was busy work.

3.1, due on 1/21

1. The first example in this half of the reading was a little confusing not in concept just in utility.  I have no idea why or where you would need the Cartesian product of these two rings but whatever.

2.  Theorem 3.1 seemed very straightforward but I'm still not sure of its utility.  Subrings made sense and seem like they should be useful.  So did subfields.  Theorem 3.2 will be useful in proving something is a subring and the proof was trivial.

3.1, due on 1/19

1. I found all the reading very straightforward and understandable. I thought the 5th example was a little strange. I did not like the fact that it said "take our word for it that Axioms 2,7, and 8 hold."

2.  We are finally getting the point!  I find it very interesting that we will be able to apply general theory to a family of systems that is covered under a group of common features.  I am very interested in linear algebra (I took 570 last semester and it is an important part of my research) so I really liked the 6th and 7th but most the 11th example.  I was reading the field definition and I was thought "matrices don't do this" but then the book showed me a subset where they do, very interesting.

2.3, due on 1/14

1.  The proofs weren't too difficult I just don't like when a proof is so wordy.  It makes it easy to get lost in the words.  This sometimes makes my proofs weaker but I think it is easier to follow.

2.  I like when things come together to make a whole that is very logical.  We are not quite there yet but I feel like we're close with congruence classes (sections 2.1-2.2) no being integrated with primes(section 1.3) and the greatest common divisor (section 1.2).  I am excited to (hopefully) soon be able to see the point of these congruence classes and their utility.

Solomon Friedberg - Boston College - Packing Primes 1/11

I thought it was interesting that I had to come out to Utah to hear someone who could possibly be my neighbor.  Being from Cambridge, BC is only about 15 minutes from my house.  

1.  The speaker introduced some new notation that was a little fuzzy because it was new, such as the pi function and Riemann zeta function.  He also used several words that I did not understand but were spoken too quickly for me to be able to write them down.

2.  I thought it was interesting that not only did he have a homework problem from the second assignment on a slide but his proof was the exact same proof that I had done.  I thought it was interesting that he hadn't seen the Pi (product) function until graduate school because I have already seen it in several instances and I still have a year of undergraduate work left.  The connection that he drew between the pi function and e was interesting.  That relationship was that the number of primes less than or equal to x (pi(x)) was approximately equal to x/lnx.

2.2, due on 1/12

1. The motivation for Theorem 2.6 was a little confusing.  I'm not sure if I totally grasped the point but I'm not too worried about it because I feel like I understand Theorem 2.6.  I also don't really like how they introduced new notation but I'm sure I will get used to it.

2.  I think it is interesting how we are now developing an algebra for congruence classes.  I am still not exactly sure why they are important but hopefully the importance will avail itself soon.  I thought it was pretty sleek how the proof of Theorem 2.6 fell out directly from Theorem 2.2.

2.1, due on 1/10

1. I thought it was strange how the notation [a] could be an infinite number of different sets, depending on the congruence modulo number.  It seems like a subscript could have been added outside of the bracket to indicate the congruence modulo number.  The introduction of the sets of all congruence classes modulo n was not very clear and it was unclear the utility of them.

2. I thought it was interesting how two congruence classes could only be disjoint or identical.  This is slightly counterintuitive but the proof was very straight forward.  I am pretty sure I have seen congruence classes before but it is still unclear to me the usefulness of them.

1.1-1.3, due on 1/7

1. At first it was difficult to follow the Euclidean Algorithm.  It was unclear to me the point of it.  After reading over it again I understood what it was.  The proof seemed fairly straightforward.

2. I think it's funny how we learned about division and prime numbers so long ago and we're still studying them.  I find it interesting how we are deeply evaluating things that are so fundamental.

Introduction, due on 1/7

Junior (Graduation), Senior (Credits)
Major: Mathematics

Post-Calculus Math Courses: Math 290, Math 334, Math 341, Math 343, Math 570.
I am taking this class to fulfill the requirements for the math major.

My most effective math teacher explained difficult concepts well, answered questions concisely, and used the book as a supplement i.e. gave examples/proofs that were not in the book so that we had two sets of examples/proofs.

My least effective teacher did not explain difficult concepts well, got flustered when answering questions, and depended solely on the book only giving examples/proofs that were in the book.

I am a member of the BYU Symphony.

I can only attend Friday's office hours because of symphony rehearsal (1-2:50 MW) but if they were at three I could attend every day.