8.5, Due April 11th 2016

1) One of the most confusing parts to me while reading was how they were trying to complete the table on page 280.  Also for me, the most difficult groups we have studied are the dihedral groups and I had a hard time following the last half of the proof of Theorem 8.32.
2)  We were basically exploring theorem 8.30 before right?  I also love the chart on 281... it is very helpful to break it down in order to understand basically what each group is isomorphic to and why.  I feel like it is a good way to conclude our exploration of groups.


For the test:

• Which topics and theorems do you think are important out of those we have studied?

As far as MOST IMPORTANT, I think it is the ones that you asked us to know how to prove on the previous midterm exams.  And just knowing the characteristics of groups and rings.

• 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.

I need to work on this last stuff that we have been studying.  I definitely feel like it is the most difficult.  I feel like my understanding is about 2 feet deep in a 20 foot pool!

 

#23 in 8.2 in the second edition.  Unless you feel like this won't help us that much with understanding this latter material.  I am not really sure what problems will help me to gain a better understanding for the final.

• What have you learned in this course? How might these things be useful to you in the future?

I think this class has really taught me how to prove something logically (identify my end goal, use the info given in the assumptions to get there, create a strategy, and go).  I am so grateful for this, because I don't fear proofs like I used to.  It makes me feel a little more comfortable and ok with the idea of theory of analysis in the fall.  When we studied rings, I felt like I better understood basic characteristics of the different sets that we work in and the reasons why things work or don't work in algebra - I feel like that is definitely something that I can use as a teacher.

8.3 and 8.4, due April 8th

1)  The most difficult thing for me in these sections were the proofs in 8.4.  I am also worried that I will be able to memorize all of these definitions.  Also, what is the importance of conjugacy being an equivalence relation in G?  How does that help us?
2) It is just cool to see all of these different theorems and I'm glad that I understand them (for the most part).  I am also glad to see a theorem from Cauchy - I have been learning about him in History of Math.  He wanted to axiomatize/generalize everything in calc.  Looks like in the meantime he helped develop abstract algebra.

8.2, April 6th

1) On the example on pg. 253 in the second edition it says that G(2) is the set of elements having orders 2^0, 2^1, 2^2, etc.  And then it says that it is the subgroup {0,3,6,9} which seem to be powers of three, not two?  I don't understand how those are supposed to be the same.  How can we know or prove if something is of maximal order.  I know that it says that elements of maximal order always exist in finite p-groups.  Is that the only way to know?
2)  I would be curious to know how long it takes people to come up with fundamental theorems, or even just theorems in general.  The proof is pretty simple for the Fundamental Theorem of Finite Abelian groups, but to even come up with a conjecture is cool.  This is probably a silly question, but are people still trying to play with different constraints and situations in order to come up with different theorems in 371?

8.1, Due April 4th

1) Is there any difference between the table of sums on pg. 246 and the direct product MXN in Z6.  They look identical, except that one is separated by an addition sign while the other is separated by a comma.
2) I guess I can't really understand what is different about this section than what we have been doing before.  Hopefully it will become more clear in class.

7.10, Due April 1

1)  If Sn is really huge, is there a way you can find all of the even permutations in Sn easily?  Will half of Sn always be even permutations and the other half odd permutations since it is isomorphic to Z2?  And for the last corollary 7.55, we know that all of the even permutations in Sn are normal, so don't you just have to show that any odd permutations are not normal?  I also couldn't follow the cases on pg. 240 too well.
2) After studying for the test, I am happy to feel like I can follow this section much more easily than the past few sections that we have been studying.  Again, I am hoping that we could talk a little about the application of these things - if there is application.  I know that often times, math is just studied for math's sake.

Due March 30, 2016

• Which topics and theorems do you think are the most important out of those we have studied?

I think the most important thing, besides the theorems, is knowing examples of all different types of groups, which is what I feel the least comfortable on.

• What kinds of questions do you expect to see on the exam?

Just like the other tests, I expect to see opportunities to complete definitions, provide examples, and prove a few things.

• 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.

**INDEX OF GROUPS, left and right cosets, order of a group and an element, lots of examples of things on the review sheet, 

7.8, Due March 28th

1)  The hardest thing for me to understand in this section are simple groups and composition factors - hopefully we can spend a little bit more time on those in class.  I actually don't even understand the notation on page 226 in the second edition.
2)  I thought it was interesting to see how much simpler it is to prove the first isomorphism theorem for groups than it is for rings, but it makes sense.

7.7, March 25th 2016

1) The hardest thing for me in this chapter are the examples, like on page 218, both of the examples, especially the one where it is showing N+(a,b) some ordered pair.  I also didn't really follow the proof for theorem 7.38.
2) It is interesting to see how quotient groups compare and contrast to quotient rings.

Rest of 7.6, Due March 23rd

1) The hardest part for me to understand was the proof of theorem 7.34.  But even then, it's not too bad.  I think it will be made much more clear in class.
2) I am hoping that in class we can see a lot more examples of this section.  And also, I am just wondering how this has application in the real world, or if it even does.  Sometimes in this class it just seems like a bunch of mathematicians got together and decided on a bunch of definitions of "abstract" objects and then played around with them based on the definitions and axioms that were pre-determined.  Is that true?  It seems somewhat like a game.

7.5, 7.6, Due March 21

1)  We sort of rushed through Lagrange's Theorem last class period, so I hope we talk about it more today and how we can use it in application.  What is the difference between a normal subgroup and a commutative subgroup.  Is there a difference?
2)  I think it is cool that we are still trying to create a list of every finite group.  I am reading the second edition - is there a lot more progress on this problem since this book was published?

7.5, Due March 18th 2016

1) At first, it was super weird for me to think that it would be ab^(-1) is in K, but now it make sense because it totally depends on the operation that we have assigned to our group.  I don't totally understand the proof for Lagrange's theorem - hopefully it will make sense in class.
2)Everything is pretty straightforward.  I think the most interesting theorem is Lagrange's theorem, although I do not yet understand the proof.

7.9, Due March 16th

1) So if I understand correctly, elements that map to themselves to not add to the length k of the cycle.  Can you compose any two groups, even if they have different cycle lengths?  Does it all depend on the amount of elements that they have?  I really don't understand the factorization of permutations.  Both theorem 7.51 and the proof is a little bit difficult.
2)  Even though I don't understand it yet, I would say the most interesting part is seeing how they factor permutations.

7.4, due pi day

1) I guess I don't why we look at an automorphism in groups but not rings.  Is it because an automorphism in rings is just trivial?  I also don't totally understand Cayley's Theorem.  What does it mean for a group G to be isomorphic to a group of permutations?  I don't see how that can be true if the cardinalities are different?
2)  Other than that everything is pretty straightforward - it is cool to see the parallels and differences between groups and rings, and it is easier to understand the proofs and ideas since we started with rings.  I wonder how it would be different if we did groups before rings.

7.3, due March 11

1)  Is the center of a group just the commutative part of the group?  When referring to subsets, subrings, or subgroups, what does proper mean?  That it isn't the whole set.
2)  This section was pretty straightforward.  I am just trying to figure out the application of groups and why we have to define things like this?

7.2, Due March 9, 2016

1)  It says that some exponent rules hold, and others do not, for example (ab)^n may not equal a^n*b^n.  But then we have theorem 7.7 which says that a^m*a^n=a^(m+n), and (a^m)^n=a^(mn).  So is it safe to say that exponent rules don't really hold when the bases are different, but they do when the bases are the same?  And also, the operation in the group could be addition, so when you say a^3, it really just means add a to itself 3 times, right?  I don't really understand theorem 7.8, and corollary 7.9.
2) I think groups are really cool because they have all of these properties that I didn't consider and they are a little bit more abstract and challenging for my mind to wrap around than rings.  It is a good stretch.

7.1, Due March 7th 2016

1) In theorem 7.2, why didn't they just say R is a group under multiplication?  Why did they create an equivalent set called U?
2) It's cool that every ring is an abelian group under addition.  Until I read the proof, I didn't fully believe it.

7.1 (to page 164), Due March 4, 2016

1)  I have heard of permutations before.  Is there any way you could go through some examples of applications of permutations?  As far as I understand, it is just a bijective function, but does it only apply to groups instead of to rings (which we are used to working with)?  With groups we just have stars because it isn't necessarily adding and multiplying like we are used to?  Are we just defining an operation?
I know that order can refer to the power in a polynomial.  Is cardinality of a set just a different meaning of the word order, or are they related in some way?
I'm not sure I totally understand what a symmetric group is.

2)  Could we consider some of the ciphers that we did in cryptography as permutations?

Due March 2, 2016

• Which topics and theorems do you think are the most important out of those we have studied?
• first isomorphism theorem
• rings/homomorphisms,isomorphisms,fields, etc.
• ideals

• What kinds of questions do you expect to see on the exam?
• Lots of definitions that we need to give (similar to the last exam), problems like the homework but not crazy hard, and proofs.

• 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.
• What is kernal again, how can we find it and why is it important?
• For an integral domain, why do we need to specify that 1 does not equal 0.  Same thing with field?
• For number 3 on the practice problems, how do we do injectivity?
• How can we know for sure that we have an irreducible polynomial?  What are the most important things to know about polynomial rings?
• HOW DO YOU PROVE AN IDEAL IS MAXIMAL?!
• Can we go over examples on the study guide?

6.3, Due February 29th

1)  For the definition of a prime ideal, I thought that whenever bc was in P (the ideal) that b is in P or c is in P.  I thought that was just a part of the absorption property of ideals?  I guess it just means that it is possible for there to be a product of two things that are not originally in P, but the product is in P?  Also, I don't fully understand why an ideal containing the 1 element is actually the entire ring.  Is there any way we could go over that a little bit in class?
2)  Maximals are a super interesting idea to me.  But how could you ever really know that you couldn't get bigger than the ideal unless you had the entire ring?

6.1 and 6.2, Due February 24th, 2016

1) In the example at the bottom of pg. 140, there is an example that talks about the cosets of I.  I'm so sorry but what does it mean to be a coset of I?  It also talks about the multiplication of cosets on pg. 146, and I just wish I understood better what exactly a coset is.
2)  It is interesting now to see these al relate to homomorphisms that we talked about before, and also how they differ.

6.1, Due February 22

1) The definition of an ideal on pg. 135 (second edition) is interesting - maybe it is not a commutative ring, but it is still possible for r*a to be in I as well as a*r.  I guess what makes an ideal different than a subring?  Does an ideal need to be a subring?  If something is countable is it considered finite?
2)  I would be interested in understanding better the usefulness of ideals; why are we interested in them?

5.3, Due February 19th

1) The idea of F[x]/p(x) having a root of p(x) is still a little strange to me if we can focus on that in class.  This is also probably silly but the way that we can know if something is irreducible is if it has no roots?  Even the extension field K that contains the root doesn't make too much sense to me.
2)  It is interesting to see that although the rest of fields modulo polynomials has pretty much mirrored the moduli that we studied before, but now for p(x) to be irreducible, there are a lot more characteristics.

5.2, February 17th 2016

1)  This might be a silly, but I don't know what it means when in the example after the proof of 5.7, when it says "we now have a ring that has Z2 as a subset."  What exactly does this mean and why is it surprising?  So surprising, that they say later, "However E does contain Z2 as an honest-to-goodness subset, without any identification."  Also, what does it mean to say without any identification?
2) Arithmetic is more interesting in this congruence-class arithmetic.  However, it still is very similar to the arithmetic that we learned in Zn.

5.1, Due February 16, 2016

1)  I like this section.  What are differences between polynomial congruence classes and just normal congruence classes in Z mod n?  The most confusing part for me was probably the examples at the end of pg. 122.
2)  I just like that all of the proofs are pretty much the same as proofs for congruence classes Z mod n.  Is the proof at the bottom of pg. 121 in the second edition rigorous?  I guess what I mean is it just seems like a general explanation.

4.5, 4.6 February 12

1) I did not really understand the proof of Eisentstein's criterion and why that woul dbe true... Is there a time when p^2 would divide the leading coefficient?  Why are the so many less irreducible in R(x) and C(x), especially when we are dealing with irrational numbers?
2) It is cool to me to think that every polynomial in R(X) is reducible, except for first degree polynomials.

4.4, Due February 10, 2016

1) For the factor theorem on pg. 102, what if you deal with something like bx-a.  Is there a theorem that tells us that a/b is a root of the polynomial?  Also, how do we know when we need to use induction for a proof (like on the proof of corollary 4.16).  How can you prove that something does not have roots?
2) I am just curious when we can switch fields to get roots, like how sometimes we say a polynomial has imaginary roots?

4.3, 2/8/16

1) So can you basically consider an associate to be something that is divisible by a certain polynomial?  And according to my understanding, it also means that if a=bu, then b=au^-1.  In the definition for irreducible polynomials, is it suggesting that some polynomials are divisible by anything?  I guess I can't really see the difference between "associates" and just being divisible by something else or reducible.
2)  Do irreducible polynomials sort of act like primes in certain ways?  Or do we sort of consider "associates" to be like a prime factorization of a polynomial?

4.2, Due February 4th 2016

1) I guess I don't really understand how to find the GCD of polynomials.  Also what exactly is a monic common divisor?
2)  It's nice to get good at all of this polynomial division and factorization for various integration problems I face in the math lab! haha

4.1, Due February 3, 2016

1)  I guess they just didn't go through a ton of the proofs of theorem 4.1 and I'm just wondering if we are going to go through those in class?  I think for the most part proofs involving polynomials aren't very difficult, per say, but more tedious.  But because there are so many variables and constants going on, I do get a little bit confused.  Also, why does it help us to know what the degree of the polynomial is?
2) I thought it was interesting that you could take any ring R (even if it is not a commutative ring) and there exists a ring P that looks like it does have commutativity, looking at the second part of theorem 4.1.

Reflections, Due January 27

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?  I spend about 1-2 hours on each assignment.  Yes they help, except on the really tricky proofs.  I need to stop procrastinating them so much so that I can really think about them sufficiently and not be pressed for time.
• What has contributed most to your learning in this class thus far?
• I would say that Lecture and Homework has contributed the most, but part of my benefit from lecture comes from me reading the chapter before-hand so I already know the questions I have and what to look for.
• What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)
• I think that I need to study the definitions/proofs a little bit every day, or even three days a week so I am not cramming so much for the test.

3.3, Due January 25 2016

1) Can you please explain what it means to be injective, surjective, and bijective?  I forgot this.  Also, I guess we can just think of "changing" the name of all the elements as putting the elements through a function?
2) How is it advantageous to us to recognize that there is an isomorphism or homomorphism from one set to the other?

3.2, January 22

1)  Is there a way we can prove that there always exists an inverse matrix (well actually one doesn't always exist) but that the determinant must not be 0 in order to find it?  I also don't understand how the 3.9 proof is proving it.
2)  I am so used to just using the fact that the numbers I work with are part of a ring so I can perform all of these operations, but I haven't thought about how we need to prove all of these theorems in the first place, before we can even consider using them for practical use.

3.1 Due January 20

1) So basically an integral domain is when there are no zero divisors?  Why do we have to specify that 1R does  not equal 0R?  And for a field to exist, why do we need to specify that when ax=1R has a solution, a does not equal 0?  I didn't really see how the proof is sufficient for Theorem 3.2.
2)  I feel a lot more confident with rings and it is cool to see how they are defined in so many different ways.

3.1, Due January 15

1) What exactly is an axiom?

So is a congruence class a ring? It seems to have the same properties.

I DO NOT GET THE ADDITION AND MULTIPLICATION TABLE on pg. 43

2) I guess I just don't really understand what makes a ring distinct or different from things that we have learned before because it has the same properties.  It sort of seemed like they put a bunch of random stuff that I have learned together in this chapter and I'm still not really sure what a ring is, but hopefully it will make more sense tomorrow in lecture.

2.3, Due January 13th

1)  When a and n are not relatively prime, I guess I don't really understand why the equation ax-b may have no solutions or several solutions in Zn.
2)  I guess it makes sense, since we use the Euclidean algorithm to find the inverse of a certain number mod(n), that we would also use that to prove that the inverse always exists.  Simple enough, but when I first saw it I wasn't exactly sure as to how we would prove it.

2.2, Due January 11th 2016

1) When it talks about the properties of multiplication in modular arithmetic, it gives names of different rules (i.e. closure for multiplication, associative multiplication, distributive laws, etc.).  However, it doesn't give a name for #11: ab=0, then a=0 or b=0.  Is there a name for that?  And why does this property hold for Z5 and fail for Z6?
2)  It's interesting to see that even exponents in Zn are the same, but I guess it makes sense since multiplication and addition are the same.

2.1, Due January 7, 2015

1) In the first paragraph when they say, "two integers a and b are equal if their difference is 0", why did they have to include: "or equivalently if their difference is a multiple of 0?"  How could it be a multiple of 0 instead of just 0?
What is disjoint again?
2) I had never really thought of thinking something is 0 if the difference is 0 but it totally makes sense: if a-b=0, then a=b.  Then since nmod(n) acts like 0 in the modulus, it would only make sense that two numbers are congruent if their difference is n (now here I can see why they would have to specify if their difference is a multiple of n.  Now I understand why we said a is congruent to b if a-b=nk.
I understand equivalence classes so much more clearly now!

1.1-1.3, Wednesday January 6

1) I didn't understand how the proof of Lemma 1.7 really proved that (a,b)=(b,r).  Especially when they are using different values for the common divisor (c in the first, and e in the second).

2) I never thought about the fact that different numbers can have multiple prime factorizations depending on the negatives.  I always just considered the case when there are only positive numbers.  I also find the way that they prove some things to be very interesting and enlightening once I stare at it for a second, but I am just hoping that I can learn to do it on my own.

Blog 1, Wednesday January 6 2016

• What is your year in school and major? 
• Super senior, Math Education
• Which post-calculus math courses have you taken?  (Use names or BYU course numbers.)
• 290, Linear Algebra, Differential Equations, Multivariable Calculus, Cryptography, Geometry proofs
• Why are you taking this class?  (Be specific.)
• Required for my major, want to become better at proofs
• Tell me about the math professor or teacher you have had who was the most and/or least effective.  What did s/he do that worked so well/poorly?
• The worst math professor that I had was very unorganized and very uninterested in the students.  Made lots of typos on the exams, and unclear in lecture.
• The best math professor that I had was very organized, kind, clear, and extremely passionate about the subject.
• Write something interesting or unique about yourself.
• I will run a half-marathon in April
• If you are unable to come to my scheduled office hours or the TA's scheduled office hours, what times would work for you?