FINAL EXAM Reflections, due April 13

I think that the important stuff to know for the final is the material on rings and groups.  I know that sounds like something to which you could reply "DUH! That's the whole course" and you might be right.  But I mean it in a more specific way.  Yeah, modular arithmetic is cool and all but I think it's more important to know about the groups and rings that are formed by the set of integers modulo some n.  I would really like all of chapter 8 to be a nice little time filler, for-your-own-enjoyment type of thing and not be included on the final.  I feel that if we've had to stick with the other section for the entire semester thus far, we might as well endure to the end and not change things up now.  I mean if we were going to break away from them, I wish we would have done that before the second midterm.  This class has taught me a lot of really cool things that I found to be very fascinating.   It helped me embrace my inner nerd. :)  I don't know how much of it I will use as a teacher but I don't plan on just forgetting it all as soon as the semester ends.  It's been fun!

8.3, due April 11

I really liked how this section stated the Sylow Theorems and then showed how to apply them, putting their proofs off until later.  Some of the times when all that information was crammed into the same section it got really confusing.  I enjoyed the plethora of examples that the authors gave and the humor they sneaked into the footnotes :) As I read, I noticed that I was picking up on the gist of the theorems and the math the involved, but they weren't sinking in very deeply.  I was getting a little turned around in the general forms of the statements of the theorems themselves, but the examples helped set things right.

8.2, due April 8

Throughout this section I could tell that I was getting more and more confused.  I was getting so wrapped up in trying to understand the new stuff that even some of  the old stuff that mention was confusing.   I thought the G(p) subgroup was pretty cool and the maximal element reminded me of something that I learned in MATH 341 last semester with least upper bounds.  I also thought that Lemma 8.8 would be very handy in some of the proofs and homework problems we would be assigned.

8.1, due April 6

One thing that confused me in this section came when they defined the coordinatewise operation on the Cartesian product G1xG2x....xGn.  I expected it to be an ordered pair with n entries.  Instead it was like to cycles each with n elements composed together.  I thought the with M={0, 3}, N={0, 2, 4}, and Z_6 was EXTREMELY fascinating!  Things weren't as awesome after that (How could you top something that cool?) and I think there's a typo in the last example.  They claim that M={1, 11} has cyclic order 4 and N={1, 2, 4, 8} has cyclic order 2.  I'm pretty sure that's supposed to be the other way around.

7.10, due April 4

This section wasn't too bad, but the proof of that first theorem was REALLY long and at times was a little hard to follow.  It was hard to keep track of what each symbol/variable was representing and what they were being used for.  One thing I learned while reading this section was that normally when you see (123) you read it "one hundred twenty-three" but this is not how it is meant to be read in cycle notation.  In cycle notation these are individual integers.   I found myself instinctively reading it the way it is meant for cycle notation instead of the normal way I would usually think about it.

7.9, due April 1

This section was pretty cool.  I found the new notation for permutations kinda fascinating.  It was really easy for me to understand.  The alternating groups were a littel tricky.  I kept wanting to do the operation from left to right but I'm pretty sure I'm supposed to do them right to left because it's still function composition.  I thought it was especially cool that you could write a single permutation as the product of two cycles.  That was really intriguing.

7.8, due March 30

Does everything in groups have a connection to something in rings?  I get so confused when they have the same notation or even names (kernel!).  I had a little bit of a hard time wrapping my head around all this in rings and now it's like I have to unwrap it and wrap it up again but this time in groups.  Another thing I noticed was that there is no Second Isomorphic Theorem For Groups.  I got a little caught up wondering why it wasn't included.  It must exist if there is a third one but was it just not important enough to make the book?  As far as the actual math goes, it all seemed straight forward until I got to the composite factors of G.  I don't even know where to start with those.

7.7, due March 28

Quotient groups, eh?  This section brought back old memories of the complexities of quotient rings.  It felt a little bit like the same song but the second verse.  When I read the first theorem, I thought to myself "isn't this something we already know?" and was able to follow the proof of it very easily.  The other theorems however were a little less obvious so me but they did make sense.  Their proofs weren't as straightforward to me but I thought I was able to get the gist of them.

7.6 (part 2), due March 25

Dr. Doud kinda spoiled the surprise for the rest of this section in class on Wednesday.  The thought that I struggled with throughout class was that just because Na=aN does not mean that na=an for some n in N.  When we went through the section on Wednesday, everything seemed to make sense.  One thing that sparked my curiosity was the theorem with five parts.  The way that I understood them is that all together they seemed to be saying that aNa^-1 = a^-1Na.  Again I got caught up thinking that this was true for all n in N.

7.6 (part 1), due March 23

The title of this section is "Normal Subgroups".  I thought these subgroups were kind of weird.  Does left congruent and left coset mean that you perform the group operation aka "multiply" on the left?  The theorems and properties of these subgroups entered my mind as the same properties of the other not normal subgroups we talked about earlier in the chapter.  I found myself wondering if these properties were the same or if there was a difference that I was somehow missing?

EXAM 2, due March 21

Oh boy! As I have started getting ready for this exam, I have started to get a little worried.  I get worried about a lot of things though, so I shouldn't be surprised.  I think the thing that has me worried the most are the functions composed of a ring modulo some ideal.  I had a little bit of a hard time understanding that when we were studying that chapter in class.  I hope that the format of this test is very similar to the format of the first exam.  I expect to see questions about groups and questions about rings of functions and ideals. 

7.5 (part 2), due March 18

The rest of this section was pretty straightforward.  I thought it was very interesting that we could tell by the order of a group which group it was isomorphic to.  I got a little lost in the proof of the theorem that said that a group of order 6 was isomorphic either to Z_6 or S_3.  The origination of ba=(a^2)b and b^2=e was not really clear, but I got really confused when compared the correspondence to S_3. I didn't really follow that.  How did they know what each element of the group mapped to in S_3?

7.5 (part 1), due March 16

This section didn't make much sense to me.  I understood the idea and the concept that I was supposed to be understanding but I just couldn't make sense of it.  I understood the definition of congruence in groups  but the section about Lagrange's Theorem went over my head, just barely out of reach.  The theorem just before Lagrange's was kinda confusing and that confusion carried over a little bit into Lagrange's Theorem too.

7.4, due March 14

HAPPY PI DAY!!!  So the way I understand it, an isomorphism of groups is an automorphism but it has all the properties of a isomorphism of rings but instead of multiplication and addition, only the group operation has to be conserved.  A homomorphism of groups of permutations is called a representation.  SO far it all seems to be making sense.  The only hang up is the symbol for isomorphism/automorphism.  The only way to tell if it is representing an isomorphism or an automorphism.  I guess that that will have to be specified in the context of the theorem or problem which makes sense.

Oh and have a great time in Italy!

7.3, due on March 11

Subgroups, centers, and cycles!  Oh my!  I feel like this section was more difficult because there was so much information in it.  The individual parts weren't too bad but all together it seemed like there was too much to chew.  The part on generators was a little hard to follow because I felt like we had already talked about the elements that generate a group, but we had actually only discussed the elements that generate a cyclic group.  That first little taste of confusion lasted through the whole theorem, so I got a little lost.  Most of the theorems seemed to be straightforward maybe because they answered some of the questions that came to my own mind.  They're proofs however weren't so straightforward.  I found myself getting hung up on notation again and trying to read the symbols instead of just interpreting their proper meanings.  I've got to stop doing that!

7.2, due March 9

This section felt very straightforward and seemed to be almost logical.  I did get caught up in the finite order of an element of a group.  I read through the first few examples thinking that they were talking about the whole group, which wasn't making much sense.   Then I went back and read them more carefully and noticed that they were talking about specific elements.  This lead me to wonder if only the groups with finite order had element that had finite order as well.  Maybe this was answered in the reading and I just missed it somehow.

7.1 (part 2), due March 7

The rest of 7.1 wasn't too bad.  I recognized a few examples from our class discussion on Friday.  I thought the dihedral groups were pretty neat.  I thought it was really interesting how abstract algebra could apply to geometry.  Some of the examples towards the end of the chapter were hard to follow the first time around but after looking at them again, I was able to get the jist of them.  I'm still a little fuzzy on the general linear groups but that will probably be ironed out in class.

7.1 (part 1), due March 4

Tis a gift to be simple! I really enjoyed this section.  It was easy to understand and it was fairly straight forward.  The only question that crossed my mind as I read was "Does * always denote the composition of two functions or is that just one interpretation of that symbol?"  From the text it seemed to just be one interpretation of the symbol which then lead me to wonder what some other interpretations might be.  I was only a little confused about waht exactly a symmetric group was.  The definition was buried in the example and I wasn't sure what properties made the group symmetric.

6.3, due March 2

Ok, so this section wasn't all that bad.   I was able to follow the bit about primes fairly well.  The part about maximals is still a little rough though.  I understand the definition...I think.  It was the proof that kinda threw me.  I started getting mixed up in the notation and then I had hard time keeping all the rings and stuff straight in my mind.  I feel like there should be a road map or a picture or something that helps us visual learners like myself keep it all organized in our minds.

6.2 (part 2), due February 28

As I finished this section, I realized that I was still having a hard time with the notation R/I.  I just don't know what it means in words. For example F[x] means the ring of polynomials with coefficients in F.  I don't know how to think of R/I in sentence form.  I also noticed that whenever you have a lowercase letter r plus an upper case letter R it seems to always be referring to a coset r + R.  I thought that might be a fiarily obvious assumption but I didn't want to teach myself a lie.  Another question I had was does K always denote the kernel or are there times that K could be just a normal ol', run of the mill ring?

6.2 (part 1), due February 25

Call me crazy, but when I hear "quotient" I think division and I don't see how these quotient rings have anything to do with division.  The notation looks like a fraction which is related to division, but I'm trying to ignore that because it's been throwing me off a little bit from the very beginning. So I guess my question is - is there any sort of division involved in quotient rings? The addition and multiplication of ideals is fairly straight forward when you pay close attention to the notation and what each bit means. As  I read through the section I almost instinctively foiled (a+I)(c+I), but then I realized that I^2 wouldn't make sense. You gotta be careful with this stuff.

6.1 (again), due February 23

I accidentally read all of the section for the last blog. So for this blog, I will relfect on the things that I read for yesterday and see if I can make sense of the things that I had a hard time understanding before.  From our lecture today, I realized that an ideal I is in fact a ring and is therefore a set. I think I understand principals and ideals generated by c_1, c_2,...., c_n a little better, altough they are still a little unfamiliar.  As I reviewed the section, I realized that I was a little shaky on the concept of a coset. Hopefully that will come out in lecture tomorrow.

6.1, due February 22

Boy, Dr. Jenkins! You weren't kidding when you said that the material was going to take one more step in the abstract direction. The concept of ideals didn't seem too different from that of an equivalence class but as I kept reading I realized that I was confused as to just waht an ideal was.  Because an ideal was initially compared to a modulo n, I immediately thought an ideal was similar to an equivalence class.  Then I read the theorem that showed that I was a subring.  Well equivalence classes aren't rings.  So, since an ideal is a subring, then it must be a set.  But I cannot shake the similarity to an equivalence class.  Am I just getting my wires crossed?

5.3, due February 18

This section was...complex. No pun intended.  I didn't really understand why if p(x) was irreducible, then it had a root. Or was it saying tht there was a root in the set of residue classes? That part went over my head.  I was able to follow the first part fairly well because I had the example to follow, but by the time we got to the last page I was confused.  I understand how x^2 + 1 has roots in the comlex numbers, but thee way I was interpreting this section, it sounded like the author was saying that it has roots in the real numbers as well.  I don't think I understand how that works.

5.2, due February 16

First of all, can I just point out how awesome it is that the author used the phrases "honest-to-goodness" and "if doing this makes you uncomfortable"?  When I read each of those lines, I laughed right out loud.  As far as the math of this section goes, it really isn't all that hard...until you get to page 127. Once I got to that point, I was lost.  The math started to get more theoretical and less concrete.  I was waiting for an example to explain the theorems but there wasn't one! It kinda left me hanging....

5.1, due February 14

This section was rather straight forward since it was so similar to the sections on modular arithmetic.  It's almost as if they go together like peanut butter and jelly.  On the last page of this section, they gave an example of a polynomial in R[x] that had an infinite number of congruence classes and then they gave an example of a congruence in Z_n[x] that had n^k distinct congruence classes.  This got me to thinking, "Is there only a finite number of congruence classes in Z_n or are there other cases that have this distinciton?"

9.4, due February 11

Ok, so the resounding question from this section was "What?" For some reason I had to read this section twice before any of it clicked.  The first time through things just didn't make sense.  I was switching equalities (especially the addition formula) in my mind so as I read the proofs everything was out of control.  I thought it was really cool (after I could make sense of it) that we used fields to prove the addition and multiplication of fractions.  I had done that in one of my education classes with diagrams and now I can do it arithmetically. Neat!

Exam 1 reflections, due February 9

On this first exam I'm expecting to see a few practical application problems and proofs of major theorems.  I imagine it being structured much like a homework assignment.  I could also see the possibility of a few definitions being on there, but I would expect these to be the less common definitions that don't necessarily apply to many theorems.  The more common definitions will show up in the proofs of the theorems themselves.  I would think that we would be asked to state and prove big theorems like the handful that have actual names (the Division Algorithm, the Fundamental Theorem of Arithmetic, etc).  The one thing that I am the least confident on is associates.  That is one thing that I need to practice and study more just to try and make sense of it before I take the exam.

4.4, due February 7

OK. So apparently everything my algebra teacher told me was a lie.  Well, maybe it wasn't a lie but it wasn't completely accurate.  I always thought that "x" only had one meaning in a function.  Actually it has two and it depends on how you are thinking about it as to which meaning "x" takes on.  If you are thinking of "x" in the context of R[x], then "x" is an element.  If you are thinking of "x" in a polynomial function, "x" takes on the standard algebraic role of the variable to define a function f: R -> R.  I felt like most of this section was clarifying all the falsities our algebra teachers taught us in high school.  The one thing that I have found that frequently gives me grief is distinguishing between f(x) and F[x].  When you read them to yourself or out loud to your imaginary friend, they sound the exact same, "f of x". I often find myself confused about which f, and now, which x I'm talking about.  Is that big F or little f? Is x the variable or is it an element? PotAtoe, potatoe? I have a feeling we can't call this one off....

4.3, due Febraury 4

I think I understand what this section is TRYING to say but I'm having a hard time understanding how it's saying it.  I get the most confused when the proof of a theorem just says to adapt the proof of a related theorem for the integers.  As far as I understand, an irreducible polynomial is analogous to a prime number and a reducible polynomial is analogous to a composite number.  I think I understand the concept of an associate of an element but I would like to see more examples of the application of associates.

4.2, due on February 2

This section was a little mind blowing.  They started by claiming that the divisibility and greatest common factors carry over to polynomials and I thought, "Cool!" But then they showed that any any constant multiple of a function was also a multiple.  That part made sense, but then we hit GCD.  How in the world can you have a GCD if you can just multiply it by a constant and get a bigger one? That's where monic polynomials come in.  A monic polynomial is a polynomial that has a leading coefficient of 1.  The gcd fo a polynomial has to be a monic polynomial.  So if you use the Division Algorithm and get a polynomial that is not monic simply multiply it by the reciprocal of it's leading coefficent and BOOM! you get a monic polynomial.  *mind blown* I'm just going to have to remind myself to remember the monic polynomial.  That's the tricky part.

4.1, due January 31

This section was pretty interesting.  It's a cool idea of writing polynomials as rings but I'm sure it's making complete sense right now.  Is x just a single value like pi or is it a set of possible values? I'm not sure this is right but I thoguht of the Division Algorithm as long division of polynomials.  That part didn't seem too hard until I got to the proof, then I got a little lost.  I think that's one of those proofs that will be easier once we have a discussion about it in class.

Course Reflection, due January 28

There are a few axioms and definitions that apply in several of the topics we have discussed.  Some of these are the Division Algorithm and the 8 Axioms that define a ring.  I can also see how the definition of homomorphism and isomorphism would be very important for throughout the course.  The fact that the class lectures follow so closely with the book really helps me make sense of this material.  The things that I have a hard time understanding from the book become much clearer during lecture.  I also like how the homework is not overly complex.  It requires a higher level of thinking but it is not overly frustrating.  I think that I can personally be better about studying the things in the book that I don't understand the first time through and even reread some of the tough points after lectures.

3.3, due January 26

I've seen the term 'isomorphism" before in some of my other math and math ed classes, but the term "homomorphism" was new to me.  I was a little surprised to see that being isomorphic depended on a bijective function.   It makes sense that it does, but my understanding of isomorphisms was minimal and I  thought two things were said to be isomorphic if they were similar properties like how triangles are similar in angle size.  A bijective function requires this to be true; it's a stronger more sophisticated way of putting it. One question I had while reading was is an isomorphism a function or something like a set or a ring?  THe book uses noth terms but I wasn't srue how to think about it.My first thought about homomorphisms was that they would be different than an isomorphism, but they aren't.  Every isomorphism is a homomorphism, but just a square is a rectangle and a rectangle is not a square, not every homomorphism is not an isomorphism.  Conviently when proving that something is an isomorphism, you prove that it is an homomorphism.

3.2, due January 24

All this talk about rings reminds me of a song: "If you liked it then you shoulda put a ring on it..." Ok now that that is out of my system, let's get back to Abstract Algebra.  This section seemed to follow quite nicley with the things we talked about in class on. The one thing that was not the easiest to understand was Theorem 3.6.  I thought that for S to be a subring of the ring R, it had to satisfy addition and multiplication under R( at least that's the way I understood what Dr. Forcade said), but in Theorem 3.6, S has to be closed under subtraction and mulitplication in S.  The rest of the section follows from simple logic.  Well I guess it's more intuition but you have to prove it first before you can actually use it.  As I read the example of (a+b)^2, I instinctively thought a^2 + 2ab + b^2, but as I read I remembered that R is not necessarily a commutative ring. That'll be a tough one to remember.

pg 48 - end of 3.1, due January 21

So, this last little bit of Seciton 3.1 wasn't all that bad.  The first example was a little confusing when I read throguh it the first time because I forgot what a Cartesian product was.  For some reason it came into my mind that a + a' was adding a congruence class and an integer together and I was baffled.  Then the lightbulb went off and I remembered what a Cartesian product was.  Everything else made sense after that.  I noitced how in our lecture on Wednesday we used Theorem 3.2 when proving or disproving that certain subsets of the integers were rings.  Reading the theorem made that process a little more clear.  In class I thought we were only allowed to do that in special cases, but here it says that we can do it for all cases where S is a subset of the ring R. That's wonderful news!

3.1 - pg 48, due January 19

This section wasn't too bad, although I was slightly confused trying to understand what the elements of a ring are. Are they congruence classes? Are they integers? Does it depend on the psecific ring you are working with?  It might seem silly but when I think "ring", I think "cirlce". Are the elements if a ring related to each other in some circular pattern?  In the book they had a lot of examples using matrices. So is a ring a matrix or is a matrix a ring? I think after a class discussion and finishing the section these things will be clearer.  I do think that it's nice that a ring is defined by the same axioms that exist in the integers and in Z_n. That was a relief. I understand those.

2.3, due January 14

This section was...interesting. I had a time following the proofs the first go around so I had to reread them, much more carefully the second go around.  I thought the prrof of Theorem 2.8 was intersting how they did (1) --> (2), then (2) --> (3), and finally (3) --> (1).  I did a proof like this in one of my MTHED classes and it was one of my favorite proofs from the course.  Using one part of a theorem to prove another part seems like a much simpler and more comprhensible of proving the theorem as a whole.  I think it's pretty cool that when n is prime you can find specific answers to modular arithmetic problems in Z_n.  When n is not prime though there could be multiple solutions, which is interesting, but I much prefer solving problems with one solution and knowing that it is right.

2.2, due January 12

Modular Arithmetic? Hooray! I thought this was one of the coolest parts of MATH290 and was so excited to use it again. My FHE brother mocked my excitement, but he just doesn't understand how cool it really is.  It's so cryptic and yet simple.  The only tricky thing is finding the equivalent congruence class.  Sometimes I forget that [0]=[n] so I'll create an extra congruence class for [n], but this can be easily solved if you create congruence classes based off of the remainder term in the Division Algorithm.  Modular arithmetic seems like it would be very useful in creating numerical codes for security or personal information because you only have n congruence classes but these classes contain every posssible integer.  Modular arithmetic is so fascinating!

"Packing Primes" by Solomon Friedberg

Dr. Friedberg's lecture was quite interesting. He showed the humorous and human side of mathematicians.  Some of the proofs he went through, especially the last two proving that there is an infinite number of primes, went over my head and were hard to follow.  As a whole however, I found the seminar to be very intriguing and intellectually fascinating.  I loved it when he shared the story about the Houston Oilers and his mathematician friend thought he meant the Houston Eulers.  It let his listeners see his true colors and helped those of us that are more dedicated to following sports than the most recent mathematical discoveries identify and connect with.  This recaptivated my interest in his presentation and I consequently got much more out of the lecture. Well done, Dr. Friedberg!

2.1, due January 10

In this reading the most challenging thing was to train my brain to think about congruence in the right way.  I kept trying to read 17=5(mod6) as 5 divides 17-6.  Once I was able to get the true meaning of that expression down the rest came fairly easily.  A small hiccup came when I got to congruence classes, but after reviewing the definition and several examples that too became clear.  When I started reading this section, I was a little excited because I remembered liking our discussion of modular arithmetic in MATH290.  I was a little rusty after a year's absence but it's slowly starting to come back.  It might sound weird but I like congruence classes.  I find them fascinating!

1.1 - 1.3, due on January 7

The hardest part of this material was following the logic that was only written in words.  I am a very visual, hands-on learner, so when the author would walk through a proof and only describe the steps he was taking instead of showing them I had a harder time following the math.  I found the many applications to division absolutley fascinating! I never realized that that was how division really worked. I knew division was repeated subtraction, but some of the things the author did just blew my mind! I really enjoy learning things like that.  I like to be able to apply the new things I am learning to things I already know. That to me is what learning math is all about.

Introduction, due on January 7

I am a junior majoring in Math Education.  I took AP Calculus in High School, so when I came to BYU I jumped right into the upper level math classes.  So far I have taken Linear Algebra, Multivariable Calculus, Differential Equations, MATH 290 and Theory of Analysis 1. I am taking this class because it is part of the requirements to receive a degree in Math Education. The best math teachers I have had are the teachers that truly care that their students learn, not just memorize, the math contained in their course.  They also structured their class in a way that allowed the students to have part in class discussions.  Something unique about me is that I love to laugh. I find at least one thing everyday that makes me laugh, most of the time it's my own clutziness or something silly that I have done. Even on my worst of days, I try to laugh about something.  I have a class during the scheduled office hours.  The hour before or after our class would work best for me. So either 10-10:50 or 12-12:50 on MWF.