6.4 of Stein's book and final study, due December 7

As far as studying for the final: I feel a little lost. It feels like we have covered a million different things and anything at all will help me to be more prepared for the final. I know that I need to know all the primality tests (by name? and procedure) and all of the various functions that have been defined throughout the semester. I'm expecting the exam to look pretty much like the others have.
As for the reading: It didn't make a lot of sense to me. It sounds like a cool idea but my brain is fried. Luckily I did a make-up blog post earlier in the semester so that I don't have to try and pinpoint the one part that was difficult for me.

6.3 of Stein's book, due December 5

First, I was a little disappointed when we started using an online book. I really don't like reading online, plus I can't take the book with me throughout the day- which is why I bought the other one. So, does B-power smooth basically mean that in the prime factorization there are no repeated primes? That's what it seemed like but I feel like if that was all it meant then they would have explained it that way. I hope that we do an example of algorithm 6.3.2 in class, because their explanation didn't really tell me how to do it. If I can understand this by the end of class today, that'd be SWEET because factorization is something we've previously had no algorithms or extra aids for.

sections 6.1 and 6.2, due December 2

I think that both of these sections were entirely over my head. The picture in the first example (above where the section actually started) made sense and I thought I understood what an elliptical curve meant, but the next picture seemed to have nothing in common with the first. It also seemed like they were using some words with a definition different than the one I am used to or maybe I just don't know how to apply those definitions to this case. (order, characteristic) The cool thing was that it said this is used in cryptosystems and that is the same thing that impossible book problem referenced so I'm interested about how/if this applies to cryptography (or I will be once I understand it).

section 5.4.2, due November 30

The RSA algorithm was difficult for me to follow. I think I understood the very basic concept, but I got lost on most of the details. It's cool that all the things we've learned about primes come into play with this algorithm. I'm wondering how pseudoprimes and other special cases affect this code. Are they wise to choose? Unwise? I'd probably be able to figure that out myself had I understood it a bit better.

5.4 through 5.4.1, due November 24

This section was really interesting and I almost just kept reading. The only part that I didn't quite understand was example 5.4.3, it didn't seem like it was really explained. It's cool to see an area where number theory is applied because there is so much of the class that is purely abstract in my mind and I sometimes can't think of a context where it might be used.

5.3.2, due November 22

First question: does the Lucas-Lehmer test always work for Mersenne primes? (It sounds like it does.) This proof is a great idea but it is far from intuitive and the proof was extremely difficult for me to follow. I know we already talked about this, but that is a huge number that we know is prime! It's cool that mathematics has so much left undiscovered. I think that's one reason I love it so much.

5.3.1, due November 21

In the example that went along with the first definition, is the book supposed to say that 25 is not a pseudoprime base 2 or 3? Because according to the definition you work mod n when testing n's primality. So it doesn't make sense to talk about using a different number there. So how many bases do we have to test to decide that n is pretty likely to be prime? Is there a general rule for this? And how would that work since we have lemma 5.3.1.1 where every time we test one more base, we're getting many more. Since we know that a Carmichael number must be divisible by three primes, it will never be prime. But, depending on how many units there are mod that number, it could have satisfied fermat's theorem for a lot of bases. Does that mean that purely looking at the number of bases that work for a number gives no good indication of primality?

5.3 up to section 5.3.1, due November 17

I had no idea the numbers that are considered to be "small" for a computer were actually so big! When they were using Fermat's theorem to see if a number could be prime, I got confused because they were looking to see if 2^11386 was 1 mod 11387, but it didn't look like they ever actually found that because they had 2^11387. It's easy to find from there but why did they stop there? At first I was all excited because someone made fermat's deterministic, but as soon as I read it I was disappointed cause it'll still take forever. This concept is a little frustration because we have proved a million ways that there are infinitely many primes and yet we don't know how to find very many of them.

5.1 and section 5.2 through the top of page 202, due November 15

I'm glad we're finally getting closer to being able to effectively test whether or not a number is prime. This section is interesting because it gives insight into how computers test for prime numbers.  The sieve method just looks like what anyone naturally does when checking for primes. I guess it's nice because you end up with a compiled list rather than doing the process for each number. The next example was a little bit confusing to me because I thought they were finding primes again but then it made more sense when I realized that they weren't trying to do that. This was about as far as my understanding extended. Theorem 5.2.1 didn't make sense to me and I think it is partly because there is the greatest integer function and a sum (i'm tired of doing sums). I can compute numbers in this formula but I don't follow the proof because I don't understand how to interpret it.

Test 2 questions, due November 11

I think the theorems that define functions are important. Also, the ones the required about 20 lemmas in order to prove. I have no idea what to expect from the exam other than a few problems that are basically computation and then 1 or 2 proofs. I need to work on understanding everything. What we've been doing recently and the proofs where we switched summations are still confusing to me. Any of the review questions will help me. For the rest of the semester I just really hope we do something that is at least accessible. It is so frustrating to be in class every day, do the readings each night, come to office hours every day, and still the material is just flying over my head. I want to understand so badly and I just don't know how.

the rest of section 4.2, due November 8

On theorem 4.2.4, I need some help with the notation and understand what it's saying. I think that Bertrand's theorem is telling us something really awesome about primes and how their density is related to where in the natural numbers we are looking (really big numbers or pretty small numbers). I think it'll be cool to talk more about the consequences/interpretations of this theorem in class. Or maybe I'm just reading too much into it.

4.2 through the middle of page 144, due November 7

I have never seen an approximation for factorial and I've never thought about one being useful because I often forget to think about these concepts to really large numbers where it would be difficult to compute them. I'm curious how good of an approximation it is. I think the following lemmas might be getting at that, but I'm a little confused about where they came from (even with the proof) and how to interpret them. I have really lost sight of the big picture, I know we talked about Chebychev's estimation last time and so of course we read the (very confusing) proof today, but the other things seem somewhat random. I know they somehow connect because we keep learning lots of lemmas in order to prove theorems, but they still seem out of place to me.

4.1 and section 4.2 through page 138, due November 4

In the first section, I could easily see that these different functions had similarities and a connection should be drawn but I didn't follow their explanation very well. I'm glad we're finally looking at the binomial theorem cause I'm hoping that this time it will finally stick! and I am excited to better understand pascal's triangle because I was introduced to it in 9th grade and didn't see it enough to gain much from it besides that.

The rest of 3.3, due November 1

First of all, I don't think I have any idea what the orthogonality relations are. I know this doesn't exactly help you know what to teach tomorrow, but I think my biggest problem right now is that we are building each day but there are always things I'm still not getting. The other thing is that the logic used in these recent proofs seems very different  to me than what we have done in the past. When 3 pages are spent on a proof and I get lost near the beginning, it is incredibly difficult for me to follow the rest of it, especially because I haven't been able to see where these are going. My favorite part of this section was that they didn't prove the last theorem. Yay!

3.3:110-115

I didn't understand when they rewrote the function by defining the piecewise function. The numerator made sense, but the denominator seemed a little out of the blue. I am also still having a hard time understanding how it is that convergence helps us to prove these concepts, it seems so unrelated. Even though this class has turned out to be very difficult for me, I suppose it is kind of a good thing. Never before have a gone to every class, done all my homework, studied lots, taken good notes and still not understand. This class is helping me so that I'll have empathy for my future students and previously I have never been in that position.

3.6, due October 28

This section seems relatively more doable than some of the other sections we have been doing lately. First of all, we have already used some of these things like multiplicative functions and we already knew what that means. One thing that was a little bit frustrating is that we were expected to be recalling the Mobius function , but luckily it's not too complicated. I do think it will be SUPER helpful to hear it explained out loud though. What does really confuse me is the inversion formula. I think what's complicated about it for me is combining so many things from different contexts.

3.3--pg 110, due October25

The first part of the reading was a little bit mind blowing. It was weird to look at it as there being 0 primes as x goes to infinity because it shows how spread about the primes get close to infinity. I think that the Dirichlet series mostly make sense although I'm not sure if I understand the underlying concept behind this proof. I mostly get that we're assuming we have finitely many primes congruent to b mod a and then we build a series around that which converges and then show that series diverges. I think I'm not quite getting the part how we build these series. With the Dirichlet characters, I am still trying to figure out what exactly that is and then what we are doing with it. I'm not sure if their notation is confusing or if I'm just looking for more than what is there to be found.

3.2.5, due October 24

As a quick sidenote, the part of class about modular form was entirely over my head...I honestly didn't grasp any of it.
I felt like some of the things in this section weren't defined very thoroughly which made it hard to follow as the section went on. Also, for theorem 3.2.5 it gave a statement and sort of made sense and then said this crazy fact came out of it without giving any explanation about how or why. And, by this point I am very much convinced that there are an infinite number of primes. I suppose it's cool that we can prove it in so many different ways and using completely different areas of mathematics to do so.

3.2.4, due on October 21

On theorem 3.2.4.1 I don't understand why showing that the product is also a sum of four squares means that we only need to prove the theorem for primes. What went on in between here? It's awesome that it somehow means we only need to prove it for primes, at least I thought it would be, but the proof gets crazy after that!

3.2.2, due on October 19

I generally understood the proofs of the lemmas in this section but I think I'm still missing how everything connects for the theorem. Another thing that was hard for me is recognizing why they were saying that certain integers were equivalent to __ mod _ . The beginning of the proof for theorem 3.2.2.2 shows us that if m=uv when u is a sum of squares and v is a sum of squares, then m is also a sum of squares. This seems so simple and I'm surprised I didn't already know it, but I have never before thought about it. This class is helping me to ask questions about things that I have previously accepted for face value and I'm learning to look for patterns and things that are always true which will help me to develop as a math teacher and also help me to aid my students in developing their problem solving skills.

Make up post (Creativity in Mathematics)

I haven't missed a blog yet, but today was a time that I could go to the math forum. So in case I miss one in the future, this is my make up blog.
One of Dr. Chamberland's points that I really liked was that many mathematicians are driven by the aesthetics of math. I feel like I can somewhat identify with this because I have looked at a page of homework that I spent hours on and thought, "man this looks good". Another point he had was that math is not useful if we don't make connections and apply it. Any area of pure math would be completely useless if there was never a connection made between it and some other area. A couple of my favorite quotes from his lecture are, "proofs legitimize intuition", "when the brain is overstimulated, learning is prevented", and "people learn better after a walk in nature than I walk through urban society". Each of these is fairly self-explanatory, but is so applicable to the way each of us studies mathematics.

3.2-3.2.1, due October 17

In the integral quadratic forms, I'm not sure what they mean by substituting integers other than (0, 0, 0, ..., 0) because I don't know where they're suggesting we substitute them. If it means for the ai's then how could it ever be something besides 0? I like theorem 3.2.1.1 because it gives us a general form we can use to find infinitely many pythagorean triples and it's not even that difficult to come up with. I hope that working with polynomials in this class works out better than it did in Abstract Algebra.

3.1.5, due October 14

This section seems a little more doable, although it is just really hard for me to learn from this book. I think most things will make sense once I get to ask questions and we fill in all the parts that the book leaves out. (by the way, I'm asking more questions in class, I hope this isn't annoying-i know we were getting low on time last class). So it seems like we can use any modulus for the lemmas 3.1.5.1-2, if this is true, that blows my mind. I think this section seems to have the most interesting properties of primes/integers that we have looked at up to this point. I'm having a hard time understanding the notation at the bottom of page 81, if we use this, I'm hoping it is explained as it's introduced.

2nd half of 3.1.4, due October 11

First, doing the polynomial division and looking at the patterns last class was really helpful and I'm surprised that the book didn't explain that part.
In the proof of corollary 3.1.4.1, how did they write fn+1=α^(n+1)-β^(n+1) ( I know I didn't write the denominators, but I just don't understand this step). Is there a lot of algebra that they left out here? Or were they using something besides the Binet Formula? I think that the Lemmas 3.1.4.2-4 are interesting because they tell us properties of the Fibonacci numbers that will help us to actually use them to prove things, and just some basic properties so that we can work with them a little bit better (and it's just cool that these things are actually true). 

3.1.4 through the first half of page 72, due October 10

One the "proof" involving the semicircle, I understood the actually math of each step and where they started from but I don't think I quite got the conclusion that they were drawing from it all. I also still don't exactly understand how the fibbonacci numbers are related to the golden ratio, it said that they're related but I never saw how. I'm interested to how the fibbonacci numbers are useful. I've seen them before and I know what they are, but it is such a simple pattern that I've never really seen how they would be useful.

3.1.3, due October 7

First, I have to admit that I am very lost with the new material that was presented on Wednesday. I think it was just too much new stuff, moving too quickly and I couldn't process it as fast as we were going. Also, I spent hours on the homework, and I worked with Ashley to talk things out and I still haven't been able to do any of the problems (one of them I did, but I feel doubtful that it's right). I talked to at least 4 other students who also feel this way. I'm not sure what if anything you can do, or maybe I just need to spend some time having this reexplained to me one on one, but I am really struggling all of a sudden. I really want to do well in this class, and I know I can, but I'm doing the readings, putting in tons of time on the homework, going to office hours, and I'm just not sure what else to do.

This book seems to skip a lot of the middle steps in proofs. For example, I was understanding the proof of theorem 3.1.3.1 but then it didn't say why we had (a^kl +1)/(a^l+1) or why knowing that a^l + 1 divides a^n+1 tells us anything. The proof that is most confusing to me is probably the one involving the GCD of Mm and Mn. The hardest part may be just trying to remember what the Mersenne numbers actually mean because I kind of have to unpack the definition at each step of the proof in order to semi follow it. Which is difficult, since they're new. I kind of like the concept of perfect numbers because I can understand it and it's interesting that anyone thought to investigate such a concept.

3.1.2, due October 4

I'm not sure if I had such a hard time understanding because I just finished taking the test when I read this section or if it was really just that difficult for me to understand. I get the concept that if 1/p diverges, there must be an infinite number of primes, that's simple, and I even understand lemma 3.1.2.1. However, when they introduce the N is where I start getting confused. The Riemann zeta function is conveniently defined so that we can easily just look at the exponent of p and see whether or not the series diverges. I can see why it is nice to have this function defined in such a way, but I don't exactly get their manipulations of it afterward.

Due on September 30

I think the most important topics are things like quadratic reciprocity and knowing how to "unpack" definitions.Also, theorems that tell you quicker ways of doing things or thinking about things. I'm hoping that the questions on the exam will be similar to 371 tests where we'll have computations, some definition type questions, 1 or no proofs. I need to work on my understanding of the Chinese Remainder Theorem, I'm still unsure of exactly how and when to use it. I would like to see more non-trivial examples of this in class.

3.1.1, due September 28

In order to prove that there are infinitely many primes, I most like the first proof, which uses p1p2...pn +1. I like it because it is concise, proves the point, and makes sense for my way of thinking. The next proof is fine, but then I don't really understand the proofs that use factorials. I don't really like have so many different proofs of the same theorem because I'm the kind of person who likes to pick what works for me and then "not worry" about the rest. I feel like if I take that approach with this section, then I'm not learning what was intended.

Jacobi Symbols, due September 26

I think this will be a great section to talk about in class because the concepts weren't overly confusing, but the lecture will help me better understand the connections. Also, from now on, when we write (a/n) (and the context makes sense) is that just a symbol for (a/p1)^e1...(a/pk)^ek? The only thing I really see the Jacobi symbol useful for right now is as the shorthand way of writing the above, or for use in definition 5.3.1.4. Is it helpful in other ways? I kind of like the whole pseudoprimes just because I feel like the name actually kind of hints at what they are.

2.6, due on September 23

First, it is important to note that I will not be at your office hours tomorrow! Not because I can't make it or because I don't want to come, but because I have already finished the homework assignment and I understand all of them! One thing I need to do is get a little clearer on my definitions, especially of quadratic residue, but also some older definitions that are make it hard to understand new things. I think lemma 2.6.1 is cool because it tells us how to quickly find or eliminate quadratic residues which, i think, are basically solutions. (maybe i'm not understanding) But if this does what I think, then it's a pretty sweet lemma. I don't really like using the "Legendre symbol" to mean quadratic reciprocity law because then it just feels like an extra two terms that I need to memorize.

2.5.2, due on September 21

I thought I understood the Chinese remainder theorem, but when they used it to find the solution to example 2.5.2.1, I wasn't sure where they got the numbers that they were multiplying together. The theorem that says if f(x) has degree higher than p, there exists a polynomial of degree less than p where the solutions to the two functions are the same is a little bit hard for me to believe. I just can't think of why this would have to be true, but if i become convinced that it is, then it's a pretty cool concept.

2.5.1, due on September 19

As soon as we start working with polynomials modulo, it becomes a bit more challenging for me. I don't feel like this section will be too difficult but I am little weary of getting into the higher degree polynomials. I like this section because things still work mostly in the ways that we've already learned. It does seem like there are more "rules" to memorize, but hopefully that is because I haven't really learned it yet and once I do, the rules will make enough sense that they won't feel memorized.

2.4.5, due on September 16

It's nice that we just talked about cyclic groups and that they were near enough to the end of abstract algebra that I remember them better than a lot of things. It's cool how we can use algebra to prove number theory as well as using number theory to prove algebra. The subjects are very interconnected if we choose to look at these relationships. I find this helpful since I did understand most of the things from abstract algebra, so when I can rely upon that knowledge, it makes it easier to learn the new things. The hardest things from this section is that I know I understood these theorems and could remember all of the "rules" at one point, and so it's frustrating when I can't remember them in as detailed of a way as I would like to.

2.4.4, due September 14

I don't think I am understanding Euler's theorem and primitive roots. I think it's because I was just beginning to understand the phi function and how to use and and I haven't quite finished wrapping my mind around it in order to utilize it in this new context. I do like the method for finding primitive roots (even if i don't completely get them) because it is concrete and I like having one solid way of doing things every now and then.

2.4.3, due on September 12

The Euler Phi function idea was a bit confusing to me, but I think that is mainly because I've only read it and this book isn't the best at explaining things. So I expect I'll understand a lot better once I've heard it explained as well. I think that will make it a lot more concrete. I still loving getting to relearn things we did in 371 because it makes them so much more solid in my mind to see them looking from the perspective of already having a fairly good grasp on them.

2.4.2, due on September 9

I feel like there was a simpler way of finding multiplicative inverses in Z modulo n, but I can't remember what it was. The euclidean algorithm and writing 1 as a linear combination of a and n makes sense, I just thought I remembered a shorter way of finding these inverses. I do feel like after reading the proof again, I better understand the process of finding multiplicative inverses and why it works. With Wilson's Theorem, I understood about the first half and then it seemed like they dropped one proof and started another so I got confused. I also didn't understand why Wilson's theorem is useful.

2.4-2.4.1, due on September 7

I'm still having a difficult time with the different language that this book uses compared with the one that we used for Abstract Algebra. I also feel like this book is less particular about things which has, so far, only confused me. Aside from that, it is interesting to look at congruence classes again and hopefully I'll quickly remember what I learned so that I can enjoy it again. I'm not sure if I quite understand what they mean by residue classes, sets of residue classes, complete residue systems...is this just different vocabulary for something we've already learned?

2.3, due on September 2

One thing I am a little confused by is the way that this book proves that any integer > 1 can be expressed as a product of primes. I seem to recall the other book using a different proof which I thought made more sense. Perhaps that is because I don't typically use induction proofs, I think I should hurry and get over that. Another thing that confused me is when they did the prime decomposition and they have 67^0 and also how they used the prime factorization in order to find the GCD and LCM. I didn't know that if an integer is not a perfect power then its root is irrational, so that's kind of neat.

Sections 2.1 and 2.2, due August 31

The lemma that said the inductive property is equivalent to the well-ordering property did not make sense to me. I think part of the problem in understanding this book is that it is a different writing style and things are explained differently than in the abstract algebra book we used. Hopefully I'll get the hang of it as time goes on. I do like that so far we're pulling from things I understood well in 371 and I think it's interesting that the basis of number theory comes from so many basic concepts.

Introduction, due on August 31

I am a junior majoring in math education, I plan to teach high school level math after I graduate. I have taken linear algebra, multivariable calculus, abstract algebra, math 290, and statistics 301. I am taking this class primarily to fulfill a requirement for my major. I'm somewhat looking forward to this class since abstract algebra was more enjoyable than I anticipated. My most effective math teacher at BYU was you because you set up early communication between you and your students, gave detailed and legible notes in class, got  your students to read ahead, and you are able to explain difficult concepts at a basic level.My least effective math teacher at BYU  moved too quickly in class, it was difficult to follow his logic/explanations, and some of the homework he assigned was overly repetitive and didn't feel useful. I love doing outdoor activities. I am only free during the last 30 mins of your office hours on Monday and Friday but if they extended until 3, it would give me more time.

Last blog! due April 13

Every named theorem I want to be sure that I know really well. Also, there are a lot of theorems that we use pretty regularly in the same general way which I feel are important to know well. I really need to work toward understand 7.10-8.3 better before the final because those were incredibly difficult for me for some reason. Any problems from the review will be helpful for me because I haven't had time to get very far into the review yet and the ones that I have done didn't go especially well. There are definitely algebra concepts that I understand better because of this class and this will definitely help me to be a more effective teacher. I've also learned a little more about self-discipline and even surprised myself by what I have been able to accomplish in this class. Obviously that's going to be helpful to me for the rest of my future.

8.3, due April 11

I feel like these theorems (once i understand/use them more) will help me to  figure out how many subgroups a group has. This would be awesome because right now I have no idea how to tell if I have found them all or not. I'm not really sure how the second sylow theorem is particularly useful (even after reading the examples). Even though I feel pretty unsure about this section, things usually come together by the time I do the homework, so i think it'll be okay.

8.2, due April 8

I've felt a little behind and confused since we started chapter 8. I feel like I understand enough to be able to do the homework but I typically have a hard time reading through the chapter and understand amidst the new notation and the logic of the proofs. Since I already feel behind, each new section is difficult for me to understand as well. It does it significantly better after the lecture, I just feel like I missed something. The Fundamental Theorem of Finite Abelian Groups is pretty cool just because it seems pretty simple/straightforward. The most difficult thing for me right now is probably keeping the notation for groups straight (especially with the direct sum vs direct product) and also applying theorems from previous, recent sections.

8.1, due April 6

What does the internal/external stuff mean? I mostly understand the example, but when we go to the general case and the proof of it, I get pretty lost in there. It's kinda cool that we have this other way of looking at the group, and that we can write out an isomorphism. I'll probably think it's cooler once I understand it more though...

7.10, due April 4

The problem with this section for me is that I didn't completely understand the A groups and what they are. So then in this section when we were proving theorems/lemmas  and extending our knowledge of them I was just a little bit confused/behind still. I'm sure once I finish the homework for the last section I'll be a little more caught up and able to understand better. It sounds like they will be really useful since they will help us to classify finite groups which we could previously only do to some extent.

7.9, due April 1

At first I really didn't like the new notation for permutations, but as I kept reading I eventually got used to it and it does seem simpler although I am still really glad that it isn't the notation we used from the beginning. I do not like/understand well the factorization of a permutation. I don't know if I just can't follow what order they are doing them in or what but for some reason when they write out the factorization I can't seem to follow what they've done or how the two sides are equal. Because of this, it was also hard for me to follow the last two proofs of this section.

7.8, due March 30

I'm a little confused since we are mixing multiple subsets with quotient groups. It's just hard for me to keep track of what group(s) various elements are part of as I go through the proofs. I do like the vocabulary for simple groups because the term seems the concept in my mind. It's pretty cool that there are only FIVE nonabelian simple groups of order less than 1000! The stuff on the last page is kind of confusing to me mostly because we again have so many subsets. Also, my favorite line from the section was, "Yes, Virginia, there is a Second Isomorphism Theorem" Haha! so random!

7.7, due March 28

Quotient groups? I had a hard enough time with quotient fields...Mostly I see the notation in the proofs and theorems and it just doesn't mean anything to me even though they defined it. I also feel like quotient groups only have notation in common with quotient fields which is very confusing to me. Also, mixing ideals into groups seems a little questionable. I suppose it's cool that we've defined quotient groups and I'm sure I'll see some interesting similarities with fields tomorrow (?).

7.6 (second half), due March 25

This section really isn't that bad, the trickiest part is probably just keeping straight when certain things hold for normal subgroups and thus remembering that sometimes they don't hold (multiplying congruence classes). I think it's cool when nonabelian subgroups end up being normal because you can't tell that they will be right off the bat. Also, the equivalence statements are pretty easy to use, and are not even THAT difficult to prove.

7.6 (through page 211), due March 23

I can't tell if I don't understand this section because it's especially difficult or if it's just because I'm so exhausted from taking the test. I do know, however, that I don't like that we have now have left congruence classes. (though at least they both have the same properties) I feel like I'll definitely be getting these mixed up and especially mix the interpretations/definitions for each. So the only good thing that I see is kind of what I said before that at least they hold the same properties of reflectivity, symmetry, transitivity and disjoint/identical property. hoorah.

Test 2 questions, due March 21

I think the most important theorems are the named ones and the other one that we are expected to be able to state and prove. It'll also be important to know all the axioms for rings and groups-how to prove something is a group. I expect the types of questions to be similar to that of the first test: one or two proofs and then computational (although I'm not entirely sure what computational problems will look like for these sections). One thing I am only starting to grasp is the idea of cosets, I can't visualize them at all which is really putting a block on my understanding. Any problem/extra explanation regarding section 9.4 would be helpful for me since that section didn't really fit with the rest of what we did around that time.

7.5 (second half), due March 18

All of the statements we can make about isomorphisms based on the order of the group is interesting (and probably useful). I think the reason I find it interesting is that I wasn't expecting them to work out like that. These isomorphisms also make it a little easier to visualize groups and wrap my head around them a little better. None of this section was overly difficult, I'm still stuck on the first half of 7.5. I didn't fully understand it I suppose and the homework was incredibly difficult for me. From the second half reading the hardest part would probably be the second half of the proof of theorem 7.3 just because it's so long and a little tricky to follow since it sort of considers lots of possibilities.

7.5 (first half), due March 16

Congruence classes of groups is really difficult for me to wrap my mind around, I know that with some practice it'll make sense, but I'm not there yet. I think just seeing/working with some different examples would clear this up for me. It's also somewhat confusing to have right cosets (and so do we have left cosets?). It is nice that we have a lot of the same terms/vocabulary just applied in new ways but that they generally work the same with some minor adjustments.

Make up blog (Dr. Vitaly Bergelson)

Honestly, most of Dr. Bergelson's lecture went straight over my head. He began by talking about a set with finite volume which was a concept I am unfamiliar with. He then went on to discuss theories that involved the densities of sets which was still something I don't understand although I quickly saw that the even numbers have a density of 1/2 and there is a pattern that follows (although I don't know what this concept means). Part of the proof he was showing involved the statement that the density of all the prime numbers is zero which broke the pattern. Anyway, he was talking about combinatorics, so I didn't necessarily expect to understand most of what he was saying. He mainly focused on Poincare's recurrence theorem, which I at least found the history of as somewhat interesting, but generally speaking I think my math vocabulary and concept understanding just wasn't high enough to really understand where he was going.

7.4, due March 14

I'm really glad the definition for isomorphism concerning groups really didn't change. And, theorem 7.18 is so cool! Even after reading the proof, I don't COMPLETELY understand why it's true but that makes it that much cooler of a theorem. I think a lot of the theorems in this section will be really useful and they're interesting because I didn't expect them to hold. What's confusing for me so far is permutations vs representations and the advantages of each.

7.3, due March 11

The proof of theorem 7.11 is one of the hardest things for me in this section because I don't have a good understanding of analyzing the order of an element using modulo or a product of 2 other numbers. I read over those theorems from the last section multiple times and they just don't make that much sense to me. I'm starting to get it, but I don't think I know it well enough to think to use it for a proof yet. I like the cyclic groups because it's an easy way to quickly come up with a subgroup and the proof of why it is a subgroup makes perfect sense! (and it's convenient that all of these are abelian) It's also really great that we've finally made the connection between the order of an element and the order of a group, it's a pretty cool connection too!

7.2, due March 9

What's great about this section is that it basically tells us that we groups generally work the way we want/expect them to. Most of the theorems that we would expect to hold true are addressed in this section and are, in fact, true. Oh joy! It's a little weird that we use the term "order" to describe the number of elements in a group and in the way it's used in this section (describing a single element) although this isn't particularly difficult, just different. Basically I think I understand this section and it won't be too bad!

7.1 (second half), due on March 7

It's cool that we can define groups using geometry transformations (that isn't something I would have thought of).  It's also cool that a ring with identity ALWAYS has at least one subset that is a group under multiplication. The hardest part to wrap my mind around  was the last example because they defined the cartesian product of real numbers and D4. Since I can't visualize this, it was hard for me to think about as a group.

7.1 (through page 164), due on March 4

The only thing I really don't understand is what a symmetric group is. The book introduced it by example right at the end of a paragraph and really didn't go into any other detail about what exactly it means. Otherwise this section pretty much made sense. The idea of a group is a little hard to grasp just because we haven't seen that many examples and the definition seems some what vague since a group is formed with just some operation.

6.3, due March 2

I like this section because it seems fairly intuitive after already considering Rings modulo primes (or irreducible polynomials). I also really like the way of thinking about primes with ideals because it makes sense! It's also cool that using this definition some surprising things come out to be prime-such as polynomials with even constant terms. I don't understand as much why you can sometimes have a quotient ring modulo a prime that is not a field, when previously that was always a field. I understand the counter example and that we can still say it is an integral domain, it just seems counter-intuitive for it to not necessarily be a field.

6.2 (second half), due February 28

I'm having a little trouble because the new stuff about kernels and the connections with homomorphisms made since in class and I understood it then, but the things they have extended it into aren't making sense to me know. For example, with theorem 6.12 I have a hard time even understanding what it is saying although the logic behind the proof makes sense. I also don't think I'm seeing all of the implications of an injective homomorphism versus a surjective homomorphism. The explanation between Theorem 6.13 and it's proof seems like it's really important and a good connection to make, but I just can't seem to understand it. Another weird thing is that I understand the examples at the end of the section just not most of what leads up to them.

6.2 (through pg 147), due February 25

So I'm pretty confused these days, but it seems to me the first part of this reading is mostly just coming up with different terms/notation to describe things we have already discussed and learned about in a way that will help us to draw new connections...? This is kinda cool I guess because we have another way to make connections and new ways to talk about ideas that we've already learned. The most confusing thing for me is the third example because I didn't understand why they were adding ( 1 + I ) + (1 + I ). I'm also still a little confused by cosets and how it is that if I is the ideal of polynomials with even constant terms, then there are two distinct cosets. I think I'm not thinking about them in the right way.

6.1 (second half), due February 23

I've having a fairly hard time thinking about mod I where I is a subring because (like the book said) it's arbitrary and I can't think about it with any specificity. The whole coset thing seemed to just suddenly appear so that was also confusing. It's cool that we have this new way of looking at congruence and new ways of defining these things but I just don't understand it very well yet.

6.1 (through pg 138), due February 22

Things are really starting to feel abstract for me now. I like the new notation and way to think about congruence classes by defining a subring. After working with them for so long, this way seems intuitive, concise, and accurate. I did get a little confused toward the end of the reading where it talked about the ideal generate by c1, c2,... because r and c were both coming from R and I didn't see the point of that ideal.

5.3, due on February 18

The paragraph that talked about extension fields was particularly confusing to me. Partly because they use K to represent the field of congruence classes and then also the application where it says that although p(x) is irreducible in F[x], it may have roots in the extension field K. I haven't yet been able to sort out what that means or why it is true.
The section that discussed introducing new number systems and explained the complex numbers, how they work, why they're valid was definitely the most interesting part of this reading. I liked the little bit of math history thrown in there, and I also thought it was interesting that you can define complex numbers in this way and bring more meaning to them. It also helped to slightly clear up my confusion on the earlier part that I read, which is a little less fuzzy now.

5.2, due on February 16

The concepts in this section aren't especially difficult since we already learned them for congruent classes of integers, however it is a lot harder to think about congruence classes of polynomials. It's especially difficult for me to see how to reduce them (which classes are equivalent) and to manipulate them in order to show this.
I think theorem 5.8 is kind of cool just because you can apply it to any field and nonconstant polynomial where you have a commutative ring with identity. I think the main reason I think that it's cool is because we already know this theorem for integers and it's convenient that it transfers over.

5.1, due on February 14

This stuff is a little bit crazy. Equivalence classes with polynomials is a little much to wrap my mind around. I'm sure I'll get used to it and eventually think it's kind of cool, but for now it just seems like a pretty strange concept. At least the most theorems and proofs just cross straight over. 

Also, now our set of possible remainders just got so much bigger! Before it was so easy to check all the possibilities when using modulo and now the possibilities are endless. oh dear.

4.5, due on February 11

The most interesting part of this reading for me was to see why my teachers taught me to factor in the way that they did. Also it was cool to see why/how our method for finding all the possible roots of a function works.
The most difficult thing for me to understand was the proof showing that the function was irreducible, the math they did in the middle of the proof was the most confusing part and I didn't completely understand their justification for everything. I like Eisenstein's criterion for proving irreducibility much better (when it is applicable of course).

Due on February 9

I think the most important theorems are the ones that have specific names. and most important topics would be general definitions that are applied a lot such as what a field is, what a ring is, etc.

I expect computational problems as well as proofs

I need to work on knowing definitions and especially theorems better so that I have them in the forefront of my mind ready to use.

4.4, due on February 7

It's kind of crazy to me that  two different polynomials can induce the same function. That seems so contrary to everything I have learned before and it's a weird idea to think about. It's also kinda of cool though that you can just substitute all possible values of x in that ring, see that for each polynomial they map to the same elements, and then you can see that you really have the same function.
It's difficult to think of x has have two different roles. The hardest part of this is that it can change whether a statement is true or false based on what the meaning of x is in that particular context. Tricky!

4.3, due on February 4

I'm having a little bit of troubling distinguishing between units and associates. I realize they are connected but mean different things, the problem is that I can't tell in what ways they're related and what things make them different. It's just kind of a confusing definition for me.
This section was pretty short so there wasn't too much interesting stuff, but I do like that they transfer the basic concept of primes into the polynomials as well. It's also helpful that they describe it as "irreducible" instead of "prime" because it makes it easier to keep things straight and I think it describes the concept better anyway.

4.2, due on February 2

I think the most interesting thing was to see specific examples of how integer rules and proofs can be adapted for F[x]. It helps a lot that the definitions, although they seem completely different in many ways, still mean the same things in F[x]. Such as the gcd still being written as a linear combination only now it's of functions instead of just numbers.
The hardest thing for me is knowing how to start the division algorithm to find the gcd of a function. I know how do use long division to divide polynomials, I just get stuck on the starting point. Or maybe it's obvious and I'm just not yet seeing the connection I need to.

4.1, due on January 31

The best part about this reading is that many of the proofs that we already did for the integers carry over and work almost exactly the same for F(x). I'm also kind of excited to get into this section because it will be proving a lot of the stuff that will be part of my what I teach eventually.
The proof of the division algorithm was pretty long and hard to follow. Partly, it's just difficult to start thinking about polynomials in such a seemingly different way. Although, it did mostly make sense and I think after I read over it again and we talk about it in class it'll get better.

due on January 28

I spend usually around 4 hours on the homework. Lecture and reading definitely prepare me for the homework.

I find it very helpful that there is required reading due before class and rather than being expected to completely know and understand what we've read, we then go over almost everything that was covered in the reading and it makes much more sense this second time though.

To make the class better I could probably schedule a little more time for the reading so that I can read even more slowly and better wrap my mind around the new concepts. Although, I already try pretty hard at this. I feel like all the questions asked by the class in lecture are very helpful and I also like that you take the time to adequately answer all of them. I really don't have too many suggestions, I think the class is going well and certainly better than I expected.

3.3, due on January 26

The most interesting part of this section was that at the beginning I couldn't understand what isomorphism meant and how it could even be useful, but then once they compared it to a function it instantly made so much more sense and I was able to get a better grasp of it. I like the definition and how it is another way of defining a function.
The most difficult thing right now is the different properties "preserved by isomorphism", and also how to prove injection and surjection. I can't keep track of which properties hold between fields and what need to hold for isomorphism, etc.

3.2, due on January 24

In this section I'm starting to have trouble seeing how everything fits together, what implies what, etc. The vocabulary for rings and fields is starting to build up and become slightly confusing. With more exposure, and after Monday's lecture, I feel like most of it will make sense. It's just getting more difficult to visualize and keep things straight in my mind.

The most interesting thing to me about this section is theorem 3.9. If it is a field, it is an integral domain and then that it is only true that if it is an integral domain it is only a field if that integral domain is finite. I think the way that works out and how they proved it is cool.

3.1, due on January 21

What I don't understand from this section is why theorem 3.1 works. I realize they omitted the proof because it is an exercise but I'm just curious how that always works.
I think theorem 3.2 is incredibly useful and I like that I can easily see exactly why it works and how these few axioms prove all of the axioms when S is a subset of R (a ring).

3.1, due on January 19

The most difficult thing for me about this section is all of the new vocabulary and concepts. Some of it has been vaguely introduced in previous classes but there are still a lot of new ideas. Keeping the different definitions separated and straight in my mind is confusing right now, but should make more sense with practice.

The most interesting part of this reading for me is that you can create a ring any way you want, defining the rules of that ring in many different ways. It is also interesting that there are so many connections between the facts we know about integers, integers modulo ( ), and so many of the other sets of numbers we typically explore.

2.3, due on January 14

The most difficult part of this section was understanding the proof of corollary 2.10 and foreseeing how it will be useful.
The most interesting part was theorem 2.8 and the general rules that always hold when n is prime.

2.2, due on January 12

The most difficult part for me is being able to think about equivalence classes given the new notation and to remember the differences in how they work versus integers.
The most interesting part is that you can define addition and multiplication for equivalence classes and prove that almost all of the properties of integers hold.

2.1, due on January 10

The most difficult part of this reading for me to understand was the second part of corollary 2.5 and the proof justifying it.
The most interesting part of the reading is the relationships between congruence and equality, the relatively simple proofs of this, and the simplicity of the two symbols similarity.

1.1-1.3, due on January 7

To me, the most difficult part of the material was understanding the Euclidean Algorithm. The process was a little bit confusing to me and I also don't entirely understand what it does.

One thing pointed out in this chapter that connected with something else from mathematics is that every integer not equal to +/-1 or 0 is the product of primes. This fact is utilized in simplifying radicals. Also, I found these sections interesting because I can see how having a better understanding and stronger foundation of these topics will help me to be a better secondary math educator because it will help me to piece together different algorithms and general math truths.

Introduction, due on January 7

I am a sophomore and a Math Education major.
I have taken Math 290 and Math 313 since calculus.
I'm taking abstract algebra class essentially because it is a required course for my major and will help me have better background knowledge of the things I eventually teach.
My most effective math teacher at BYU was Dr. Gerson because she moved at a good pace and clearly explained all of the topics. My least effective math teacher at BYU was Dr. Conor because he moved too quickly in lecture, it was hard to follow his logic/explanations, and some of the homework he assigned was over repetitive and didn't feel useful.
I love snowboarding and am the 3rd oldest of 8 kids in my family.
Your office hours work great for me! I feel like this will be a challenging class, but you seem like a great teacher!