8.5 and final review, due on December 7

Reading:
1. The proof of Theorem 8.33 was a bit confusing since it mentioned so many results that weren't from this section.

2. I really liked seeing the table of groups extended to 15.  It's interesting to see T show up in there; it's a group we have never heard of before.


Exam preparation:
I think that being familiar with lots of examples will help more than anything else for the test.  I expect to see some of the group classification theorems on the final, since we haven't been tested on those yet.  I need to go over those a few times to make sure I know what they say, especially since some of them aren't intuitive.

One thing that I've learned already is that group theory makes recognizing patterns much easier.  If we have an unknown group, then knowing so much about the groups we have classified makes it easier to identify what the unknown group is isomorphic to, which gives us a lot of information about its structure quickly.  Abstract algebra will also help with my future math classes, since I will be doing grad school in math.

8.2, due on December 2

1. It was difficult to follow the proof of Lemma 8.6.  Could we go over that proof in class?

2. The main result of this chapter was pretty interesting.  It definitely makes working with abelian groups a lot simpler; we can classify them very quickly instead of starting from scratch with each.

8.1, due on November 30

1. The hardest part for me to understand is distinguishing between the first form of the direct product and the standard Cartesian product.  Is there any difference between the two?

2. I've seen direct sums before in vector spaces, and this seems to be a similar concept. We're decomposing G into normal subgroups (which are analogous to vector subspaces) that can be added together with no overlap to create G.

Test review, due on November 28

I think the most important theorems are the ones that allow us to manipulate groups more easily, such as Lagrange's Theorem, the Sudoku Theorem, and the fact that subgroups of a cyclic group are cyclic.

I expect to see questions on the exam that are similar to the homework problems but are things we haven't seen before.  I feel like the best way to test our knowledge is to have proofs on the test that are unfamiliar but follow easily from the theorems we have learned.

I need to practice proving the theorems on the practice exam.  I learned the proof of the First Isomorphism Theorem for the last test, but I haven't yet tried to reconstruct it with group notation instead of ideals, and I need to practice the others.

7.10, due on November 22

1. I had a hard time following all the permutation notation in the section.  When reading it's much easier to assume they're doing the steps correctly than it is to check every detail.

2.  The main theorem in this section was quite interesting, especially given the commentary on group classification in the previous section.  This section gives us an infinite set of easily-described finite simple groups, which will work well for examples in the future.

7.8, due on November 21

1. The hardest part of this section was Theorems 7.43 and 7.44.  Everything earlier in the section is similar to things we did with rings and ideals (and vector spaces), so it's not difficult to apply it to groups.  Once we start talking about quotient groups of quotient groups, though, it gets much harder to tell what's going on.

2. I thought the section on simple groups was really interesting.  It was a cool insight into some of the work there still is to do in group theory. (Also, it was much easier to understand than the rest of the section.)

7.7, due on November 18

1. The hardest part for me was keeping the elements of the group distinct in my mind from the elements of the quotient group.  It's mostly a lot of notation to keep track of.  Also, I don't really understand the example about Q/Z.  It's hard for me to figure out just what those cosets look like.

2.  It's interesting to me to see just how quickly we are learning about groups compared to rings.  I guess it makes sense because it's later in the semester, but it feels like there is a lot more packed into each section than there was in the first few chapters.

7.6 part 2

1. Not much of this was difficult, especially since we went over some of it in class.  The hardest part was probably following the proof of the interconnecting parts of Theorem 7.34.

2. It's interesting to me that even though Na = aN does not imply that na = an, that doesn't end up mattering for quotient groups.  Having the entire subgroup commute ends up being enough.

7.5 - 7.6, due on November 14

1. I think the hardest part for me is still getting  used to the multiplicative notation for cosets.  I'm sure it will come more naturally after I've done some homework on it.

2. I think it's interesting that the lack of commutativity is what makes quotient groups difficult to define, but that we don't actually need commutativity for each pair of elements in the group to make them work, only for the left and right congruence classes.

7.5, due on November 11

1. The multiplicative notation for cosets will take a bit of getting used to, since cosets in rings and vector spaces use addition as the operation.  The fact that right cosets aren't necessarily left cosets is also new, although it makes sense given that our operation may not be commutative.

2. I thought Lagrange's Theorem was quite interesting.  It's the kind of result that isn't intuitive without looking at examples.  This theorem will definitely make it easier to classify finite groups, especially with Corollary 7.27.

7.9, due on November 9

1. Theorem 7.51 in the second edition was rather confusing since we haven't learned about anything it uses in the proof.  We also haven't learned about normal subgroups or group indices, so the statement of the theorem didn't make any sense do me.  I'm guessing that's just a consequence of the difference in editions;  basically what the theorem says to me is that An is a subgroup of Sn, and not much more.

2. This section was pretty interesting.  I have played around with rearrangements before and noticed that any permutation can be achieved by just swapping two elements at a time, and I've also noticed the even-odd swap pattern.  It's nice to see both of those as theorems in the section.

7.4, due on November 7

1. Cayley's Theorem was a bit difficult to understand at first, especially since it isn't obvious from the statement of the theorem how you would go about proving it. It also relies on an earlier theorem  (7.19) that itself isn't intuitive.

2. It was interesting to see group representations come up in this section, since I am signed up for Group Representation Theory for next semester.  (I hope that class can stay scheduled!)  It was also interesting to see Z and Z mod n show up again in an abstract setting, in the characterization of cyclic groups.

4.3, due on November 4

1.  Mostly the difficult part of this section was the number of new ideas it introduced.  Most of them are things we've never seen before, which makes it harder to keep track of them all.  I read the section this morning, and I had completely forgotten about the center of a group by the time I finished reading about cyclic groups and generators.

2. I think it's really interesting that we only need to check one axiom for finite subgroups.  Subrings and ideals (and subspaces of vector spaces)
require two, even in finite rings.

7.2, due on November 2

1. The most difficult part for me was getting used to the multiplicative notation, especially the exponents.  It made it a bit difficult to follow the proofs of Theorems 7.8 and Corollary 7.9.

2. It was interesting to see how modular arithmetic pops up all over the place.  Whenever there were any calculations in the book with the order of an element of the group, they was almost always involved congruence in Z mod something.

Test review, due October 26, 2016

The most important theorem we've learned is probably the First Isomorphism Theorem (since you said so in class :) ).  In general, I think ideals are the most important thing we've studied, since they include almost all of the other things we've learned.

I expect to see problems on the test that are fairly similar to the ones we've done on the homework, but not identical.  I like being asked to come up with something new rather than redoing problems I've already done.

I could use some more review of prime and maximal ideals, and some more work with ideals in general.  This is both more recent and more difficult than the other things we've done in class.

6.3, due on October 24

1. The idea of a maximal ideal was a bit tricky for me, since we've never covered anything similar before in this class.  It was easier to see how prime ideals connect to the ideas of primes and irreducibles.

2. It was interesting to me that I being prime no longer implies that F[x]/I is a field.  That's something that our intuition from previous chapters definitely wouldn't have predicted.

6.2, due on October 21

1. The most difficult part of this section for me was understanding the functions associated with the proof and examples of the First Isomorphism Theorem.  The functions themselves aren't too hard, but they are obscured by a lot of notation, especially in the proof.

2. We learned about the First Isomorphism Theorem and the canonical epimorphism for vector spaces in Math 344 a month ago!  It's amazing how many similar ideas are popping up in both rings and vector spaces.

6.2, due on October 19

1. I know that "nothing" isn't an acceptable answer for this part of the question, but there wasn't really anything difficult for me about this material.  Cosets were difficult for me when I first learned about them in Math 344, but they behave exactly the same way with respect to ideals as they do to vector subspaces, and the notation is the same.

2. It was interesting to see how the concepts and theorems we have been using for principal ideals in Z and F[x] generalize nicely to quotient spaces with arbitrary ideals.  It's always cool to see connections like that between different mathematical ideas.

6.1, due on October 17

1. I don't understand why R must have an identity for Theorems 6.2 and 6.3.  Wouldn't the proof still work if R was a commutative ring without identity?  Also, are there any good examples of ideals that aren't finitely generated?  Never mind, I just realized that the second example on page 6 is an ideal that's not finitely generated.

2. We learned about cosets of vector subspaces in Math 344 (one of the ACME classes), and Dr. Grant mentioned that those behave just like cosets in a ring.  It's really cool to finally be learning about those, especially after having the background to understand them better.

5.3, due on October 14

1. I didn't understand at first why Corollary 5.12 holds for polynomials that are the products of nonlinear irreducible polynomials, but then I read the proof again more carefully and understood it the second time.

2. I thought it was really interesting that for every polynomial p(x) over a field, there is an extension field where p(x) has a root.  It was also really cool to see the definition of the complex numbers in terms of the extension field modulo x^2 + 1.  That was something I had never considered before.

5.1, due on October 10

1. The hardest part for me was remembering that associates aren't essentially identical in congruence classes the way they are in factoring.  If we're dealing with the congruence class [ (7/9)x + 2 ], we can't multiply by 9 to clear the fractions or multiply by 9/7 to make it monic, since those would be different congruence classes.

2.  I thought it was really cool how easily results from Z mod n translate to F[x] / (p(x)).  Only one of the theorems (Corollary 5.5) had to make any changes, and they were very minor.

4.5 and 4.6, due on October 7

1.Eisenstein's Criterion was a bit confusing at first, especially the seemingly arbitrary condition that p^2 does not divide a_0.  It makes sense after reading the proof, though.

2.I've seen most of the results in these sections before, so a lot of our reading was familiar.  I thought Lemma 4.21 was quite interesting, since it's not immediately obvious that you can't have the coefficients that are divisible by p be split up between g(x) and h(x).

4.4, due on October 5

1. The hardest thing for me is probably keeping straight the actual polynomial and the function induced by the polynomial, especially when both use the same symbol.  The paragraph on true and false statements in particular took a little while to wrap my brain around.

2. It was really interesting to see concepts from high school algebra showing up in an abstract algebra class.  Corollary 4.16 is one good example of this; it's a generalization of the same theorem about roots of polynomials in R.

4.3, due on October 3

1. The most difficult thing for me from this section was following the proof of Theorem 4.11.  Even that wasn't very hard; it just took a careful reading.  Most of the material in this chapter is familiar.

2. It's interesting to see how most of the theorems are similar to theorems about primes in Z.  I hadn't really thought about common properties between prime numbers and irreducible polynomials before.

4.2, due on September 30

1. The hardest part for me is probably keeping track of which polynomials are which in the proofs. That's mostly because the notation for polynomials is a bit more complicated in general than for numbers.

2.  I thought it was interesting that there is always only one gcd for any two polynomials, even though the gcd is simply the common divisor with the highest degree.  It's not immediately obvious that there is only one monic polynomial that satisfies this definition, but it ends up being true.

4.1, due on September 28

1. The most difficult thing for me is thinking of x as a specific element of the ring rather than a variable.  It makes sense logically, but I'm just not used to it yet.  The example with pi does help with that.

2. I thought it was really cool how well the division algorithm in F[x] correlates with the division algorithm in Z.  Both are fairly simple theorems that are extremely useful later.  I haven't read ahead in the book, but I've heard hints of some interesting ideas that are coming - cosets of functions modulo a polynomial, etc.  I'm looking forward to it.

Test preparation, due on September 26

I think the most important theorems out of the ones we've studied are:
-Every field is an integral domain (and the definitions that say that both of those are commutative and have identity)
-The division algorithm
-Properties of rings that don't come from the definition
-Properties of modular arithmetic

I expect to see questions on the exam that are fairly similar to the homework - proofs that are fairly straightforward but require an understanding of the concepts.

Before the exam, I would like to better understand the proof that every finite integral domain is a field (and the related proof that cancellation is valid in an integral domain).  I will need to review those before taking the test.  I would also like to understand quotient fields of integral domains better, but that will probably come as I do the homework due Monday.  (I won't be in class on Monday, but these are things I need to study on my own.)

Class feedback, due on September 21

I usually spend about an hour on each homework assignment.  Some of the easier ones have taken less time than that, but I haven't spent much more than an hour on any of them.  Proofs in general tend to come naturally to me, and that is what most of the homework has been.  Also, I've seen most of the concepts in the first two chapters before.

Probably the thing that has contributed most to my learning in this class has been doing the readings before coming to class.  That way, when I hear a concept in lecture it's familiar to me, and if there is a proof I didn't understand I can pay much closer attention to it to understand it better.

3.3, due on September 19

1. I had a harder time understanding a homomorphism than an isomorphism.  It still feels like just an abstract property to me rather than a description of the function.  Isomorphism makes much more sense; it's a way to tell if two rings are essentially the same.

2. We have been learning about isomorphic vector spaces in one of the ACME classes (Math 344), so the concept of an isomorphism is vary familiar.  For rings, however, there are many more nice properties than we covered for vector spaces, such as the fact that isomorphisms preserve commutativity and units.

3.2, due on September 16

1. The term "unit" was a bit confusing to me at first.  Normally when I've seen it before it means something that is equal to 1 in some way, such as the unit circle or a unit vector, but here it means something that has a multiplicative inverse (so that, for example, every nonzero element of R is a unit).

2.  I thought it was really cool that every finite integral domain is a field.  That's a result I never would have expected, and it's a bit mind-boggling to think that Axiom 11 implies Axiom 12 in the finite case but not the infinite one.

3.1 part 2, due on September 14

1. The most difficult part was probably keeping the three plus signs straight in the Cartesian product example.  The example itself wasn't that difficult, though; it makes sense that the Cartesian product of two rings is itself a ring.

2. The conditions required for a subring reminded me of the conditions required to show that a subset of a vector space is a subspace (from Math 313 and my ACME classes).  In both cases, a harder problem becomes easier when it's inside something that already satisfies the conditions.

3.1 part 1, due on September 12

1.  One thing that was difficult for me was understanding integral domains.  Are there any fairly simple examples of integral domains that aren't fields?  There were good examples of everything else in the textbook (noncommutative ring with identity, commutative ring without identity, noncommutative ring without identity, and field), but not of integral domains.

(After I wrote this, I realized that Z itself is an integral domain that's not a field.  Silly me.)

2. It was really interesting to try to wrap my brain around the example on pages 43-44.  It's the kind of example that you wouldn't expect to have any nice properties just from looking at the operations, but it does.

2.3, due on September 9

1. The most difficult part for me was understanding what Corollary 2.10 was saying.  I eventually got it, but I had to think about it for a little while, especially to understand why (a,n) = 1 is required but (b,n) = 1 is not.  It makes sense now, though.

2. It was interesting to see all the nice properties of the integers modulo a prime.  I thought it was really cool that every number except 0 has a multiplicative inverse, turning it into a field.

2.2, due on September 7

1.  Probably the most difficult part for me was the "new notation" section, especially distinguishing between exponents in the normal integers and numbers in the Z_n ring.  Even that wasn't very hard, though; I've seen all of these things before.

2.  I really liked the non-example they gave to show why we had to check our definitions of addition and multiplication for modular arithmetic.  It goes to show that we really do need to prove everything we do in mathematics.

2.1, due on September 2

1. The most difficult part to understand was probably the proof of Corollary 2.5.  I had always just assumed that all the equivalence classes were unique, but I guess that needed to be proved just like everything else we learn in this class.  Even that proof wasn't too hard; this section's material doesn't seem to be very difficult.

2.  This section's material is mostly things I have learned before, in Math 290 and in other books.  I'm looking forward to learning about other examples of rings.

1.1-1.3, due on August 30

1. The part that was the most difficult for me to understand was the chapter on greatest common factors.  I know what they are, and I have used the Euclidean algorithm before to find them, but Lemma 1.7 is something I haven't seen before, and it took a careful reading to understand what it was saying and how it applied.

2. I thought the well-ordering axiom and its uses were really cool.  It's similar to the Axiom of Completeness in Math 341 for the real numbers; it's something that intuitively makes sense but that I didn't really think about until we used it to prove lots of other theorems.  I also enjoyed reading about primes; I have seen all of the results in that chapter before, but it was good to see them again.

Introduction, due on August 30

I am a junior majoring in ACME.  I have taken Math 290, 313, 314, 334, and 341, and I have taken Math 391R multiple times.  I am taking this class because I am doing Mathematical Theory for my ACME concentration, and professors and students I have talked to have recommended this class as a basic course for mathematical theory.  It is also a prerequisite for many of the other advanced math classes and for lots of research opportunities.

The professor I had who was the most effective was Dr. Kening Lu, who taught Math 341 this past winter semester.  He was a fun professor, taught the material in a way that was easy to understand, and answered questions as needed.  Something interesting about myself is that I love to sing, and I'm currently in the process of auditioning for Men's Chorus right now.  Your office hours don't work with my schedule, but the TA's hours do, so that should be fine.