Final review questions, due on April 13

Questions: Could we do some examples of quotient rings and and Sylow stuff? Like your question 1 on the midterms kind of stuff. I'd really like to work Sylow stuff more. Doing related proofs (like number 5 on midterm 2) about rings and groups would be good too.

8.5, due on April 10

Difficult: I don't know how important class equations are, but if they are quite important could you explain them a bit in class? There were 2 lines I didn't understand in the proof of Thm 8.33.

Reflective: It's kinda nice that the book computes symmetry groups with just r and d like we've been doing in class anyway.

8.4, due on April 8

Difficult: I didn't completly understand Theorem 8.21 or the proofs of the 2nd and 3rd Sylow theorems.

Reflective: Conjugacy reminds me of similar matrices in linear algebra. Could we do more examples of those in class?

8.3, due on April 6

Difficult: So everything in this section makes sense. The problem isn't that the concepts are difficult, it's that there are so many of them. It's hard to commit it all to memory (for the test) and get an overall idea of what is going on.

Reflective: Again the concepts by themselves make sense, but could you do and example or two (like those in the book) that give an overarching idea of what is going on. The more examples, the better.

8.2, due on April 3

Difficult: So I hope we don't have to reprove the theorems, because I really didn't understand most of the proofs, but I think I got what the theorems were saying.

Reflective: So basically finite abelian groups are direct sums of cyclic groups and each has a unique set of elementary divisors. Two finite abelian groups are only isomorphic if they have the same elementary diviors. Right?

8.1, due on April 1

Difficult: So this section wasn't too bad. I would just like to see some more examples of groups that are direct products of some normal subgroups and go throught the material again.

Reflective: This makes groups seem more like numbers since they have factors. It would be kind of like the number 12 is a direct product of 1,2,3,4,6,12 where 1,2,3,4,6,12 are all normal subgroups. But it is a little different.

7.10, due on March 30

Difficult: So how in the proof for thm 7.52 does it come up with the cases, why doesn't 4 work, and where did it use Lemma 7.53?And in Cor 7.55, how does that proof go? This is tricky stuff!

Reflective: So A(sub)n is usually simple. I guess that is important so it is isomorphic to the integers mod p.

7.9, due on March 27

Difficult: Maybe it's cause it's late and I'm tired, but this is tricky to me. I'm still trying to get how (243) (1243) = (1423).

Reflective: This acts a lot like functions in a lot of ways, but is totally unlike anything I've ever done in a lot of ways.

I need to work on:

1. computations in D4
2. Right congruence
3. Left congruence
4. Cosets and what they "look like"
5. Quotient groups and what they "look like"
6. Knowing content of Thm 7.8

Questions for class:

1. a sample problem of application (kind of like #5 on last test. You know how we have to apply a thm to something we've never seen before. I'm not asking for the actual problem that will be on the test, but maybe something similar)
2. examples of quotient groups
3. examples of cosets

7.8, due on March 23

Difficult: The proof for part 3 on Thm 7.44 is still a little confusing. Also, factors in the integers you multiply to get that number. Do you compose composition factors to get a group?

Reflective: So multiplication factors are to integers as composition factors are to groups.

7.7, due on March 20

Difficult: It's a little tough to see what every quotient group "looks like." I'd like to go over the first example, D/N = Nr and Nv, again.

Reflective: This is a bit easier than quotient rings, but maybe that's because it sort of a second time through.

7.6, due on March 18

Difficult: I would like to cover both left and right congruence again and cover the definition of normal with examples just so it all sinks in.

Reflective: This is definitely different than rings.

7.5 part II, due on March 16

Note: So I read all of 7.5 for last blog for some reason so my old blog still stands.

Difficult: Corollary 7.27 makes sense when you take the time to take it apart. But I don't feel like I have the overall intuitive idea of what is going on because I'm still trying to memorize Theorem 7.8 and 7.14 and 7.18.

Reflective: Congruence is definitely a little different in groups than in rings and integers.

Addition: I will add that while the last 2 theorems make sense, they are long proofs! So it's a little hard to remember what you were even proving by the time you get to the end of the proof.

7.5 part I, due on March 13

Difficult: Corollary 7.27 makes sense when you take the time to take it apart. But I don't feel like I have the overall intuitive idea of what is going on because I'm still trying to memorize Theorem 7.8 and 7.14 and 7.18.

Reflective: Congruence is definitely a little different in groups than in rings and integers.

7.4, due on March 11

Difficult: I had a hard time following the proof for Theorem 7.20. It would be good to do that in class mayber.

Reflective: I noticed that Theorem 7.19 is a lot like Theorem 3.12. That's cool because it has a lot of nice properties.

7.3, due on March 9

Difficult: This is sort of a lot of new information. A little hard to take it all in. I'd like to see lots of cyclic subgroups and generated subgroups examples.

Reflective: Again, lots of information. The generated subgroups remind me of rings generated by two elements, like the polynomials with even constant terms generated by (x, 2). The cyclic supgroups seem completely new to me.

7.2, due on March 6

Difficult: The proofs of theorems 7.8 and 7.9 are a bit tricky. Also, the theorems themselves make perfect sense now, but I can them getting tricky if they are generalized in future sections.

Reflective: (ab) = b-1a-1 reminds me of something I think from linear algebra, but I can't seem to remember what.

7.1 part II, due on March 4

Difficult: One question. Why is a nonzero group under multiplication never a group?

Reflective: This section is nice because it gives us a lot of nice properties in groups that we are used to in rings. The whole group of shapes thing is kinda wierd, but cool. It reminds be of a game we played in elementary where we had to figure out what a piece of paper looked liked afer being folded, or rotated, or had a hole cut out and then was unfolded. Never thought of it as a function before though.

7.1 part I, due on March 2

Difficult: This section was pretty straight forward. But it's probably going to get abstract really fast. So I should make sure I get the definitions solid right now.

Reflective: I remember doing Permutations and Combinations in Algebra II. I remember that it made sence then, but I don't know remember much about it. I wonder how similar groups are to rings.

9.4, due on February 27

Difficult: If R = integers and F = rationals, then what is Rstar in Lemma 9.29? Because it seems like Rstar = integers. And what is K in Thm 9.31? The real numbers.

Reflective: I like this section! I've never thought of the rationals this way before and its kinda cool and makes sense. This section isn't too abstract.

Review QA, due on February 25

The five theorems listed in class I need to know with their proofs. Also, Cor 6.16(when max ideals are prime), thm 6.11(K=0 iff f is inj.), thm 4.11 (equivalent statements to p(x) is irreducible in a field), and 6.15 (maximal ideals and fields)and all the thms that generalize congruence classes and their properties to other rings and quotient rings other than the integers mod p and fields mod polynomials.

I need to know the definitions: maximal, prime, kernel, natural homomorphism, quotient ring, ideal, principal ideal,cosets, and greatest commone divisors . Also I need to remember that a+I is in I iff a+I = 0+I in R/I. And a-b is in K iff a is congruent to b modK. And I need to know when a polynomial is irreducible. I need to get better at figuring out what cosets "look like".

Questions:
Can we do some examples of what cosets "look like" in tricky quotient rings?
Can we define as a class: quotient rings, cosets, ideals?
Can we cover some examples of prime ideals that are not maximal?

6.3, due on February 23

Difficult: It all makes sense if I take it nice and slow, but I feel like it adds to the ton of information I'm already trying to remember. I'd like to go over the proof for Thm 6.15 again.

Reflective: So let me recap. An ideal P is prime iff R/P is an integral domain, R/P is a field iff P is a maximal ideal and a commutative ring R with identity's maximal ideals are all primes.

6.2 part II, due on February 20

Difficult: I'm not understanding entirely how Thm 6.11 works, about the kernel being 0 iff the homomorphism is injective. The rest I have a vague understanding of.

Reflective: Very abstract, but I think pictures of drawing pictures of the sets' elements mapping to their image helps a little bit.

6.2 part I, due on February 18

Difficult: So I know that quotient rings are analogous to the integers mod something or F[x]/(p(x)). But what is the difference? I am still trying to figure out exactly what it is. What does the book mean when it says: "One sometimes speaks of factoring out the ideal I to obtain the quotient ring R/I."?


Reflective: This is abstract stuff. Hopefully I catch up with it in class.

6.1, due on February 17

Difficult: I'm still a little confused about cosets, but the rest makes sense. So the coset of a+I is where a is the remainder I guess. The last example in the section just doesn't make sense. Weird notation.

Reflective: So a+I is like [a] and modn is like modI where I is a subring and ideal of some R. I think I got it.

5.3, due on February 13

Difficult: This section was very abstract. (kinda like the title of the course suggests it will be...funny.) Theorem 5.11 I don't entirely understand. Why is it important? And the proof doesn't make sense yet.

Reflective: Theorem 5.10 makes sense because it is analogous to the integers mod a prime. But 2 of the 3 proofs I don't really follow. Are groups more complicatied or less complicated than rings?

5.2, due on February 11

Difficult: Why do the theorems and definition exclude constant polynomials? And what is so nice about every field containing an isomorphic subring?

Rwflective: The chapters are getting more and more abstract. This makes it a little tricky.

5.1, due on February 9

Difficult: I still don't get the proof forCor. 5.5 very well and I need to prcatice this stuff. It's a little weird.

Reflective: It's nice that mods and congruence classes are similar with polynomials as they are with integers.

4.5and 4.6, due on February 6

Difficult: All of the thereoms, lemmas, and corollaries made sense intuitively, but it took a lot of brain energy to follow the proofs. I don't have them down solid; just the basic idea.

Reflective: These are interesting, helpful patterns to know. I especially like the Einstein's Criterion and the Rational Root Test.

4.4, due on February 4

Difficult: This chapter seemed to be a bit more abstract than the other chapters. And "induced" threw me for a while. So I guess it means that a polynomial acts as a function and so certain things make sense like setting it equal to 0.

Reflective: It's good to see the Remainder and Factor Theorems written out because I have known them, but I always forget about them.

4.2 and 4.3, due on February 2

Difficult: The definition of an associate was a little tricky at first. I'd like to see some more examples. The rest wasn't too bad.

Reflective: This section makes a lot of sense considering it's just applying the rules we are used to using to polynomials, which we are pretty used to any way. It's just a matter of knowing all the detail of the definitions.

4.1, due on January 30

Difficult: The proof for the Division Algorithm is hard to follow. The example makes sense, but the actual proof is a little fuzzy. Also, I'm having a hard time understanding the difference between indeterminates and variables. They seem like one and the same thing to me.

Reflective: I thought it was interesting how it defines the polynomials with the coefficients as elements of a subring of the entire ring where the indeterminates are.

P.S. - Thanks for no homework due on Friday. That really helps with the test.

Test Preparation, due on January 28

I will need to know the Well Ordering Axiom, The Division Algorithm, Euclidean Algorithm, The Fundamental Theorem of Arithmetic, the theorem which says that the gcd of is the smallest positive integer that can be written as a linear combination of two integers and maybe their proofs (?). And I will need to be able to use all other theorems to work problems or write proofs.

Also I will need to know the definitions of mods, concruence classes, rings, commutative rings with or without identity, integral domains, fields, subfields, units, zero divisors, isomorphisms, homomorphisms, bijective functions, and properties of the integers mod a prime number.

I think I got it all...I'll find out tomorrow.

Catch up questions, due on January 26

Homework: I have spent 1 -2 hours on each homework assignment and yes, the lectures and reading do prepare me for it. Actually doing homework problems contributes the most to my learning.

Likes: I like the material because the proofs are interesting puzzles to me. And your office hours are very convienient- thank you. I think all I lack is working with what we have learned even more so it sinks in. But I'll do that when I study for the test.

Dislikes: The only area for improvement I think is the homework/reading schedule. I like working on homework that we have just read and lectured on. Doing homework for 3.1 after attending the 3.2 lecture was a little tricky. But I realize this happened because 3.1 was so long.

3.3, due on January 23

Difficult: The material makes sense but there is a lot of it and it's going to take some practice to get it down.

Reflective: Now I know why you pointed out to Dani and I that the homework problem using r, s, and t was really the "same" ring as the integers mod 3. So they were isomorphic. And homomorphic. That makes sense.

3.2, due on January 21

Difficult: I am still trying to wrap my head around units and zero divisors. Am I right in thinking that the integers have no zero divisors?

Reflective: When I learned my math facts in elementary school, I had no clue that what I was doing was so abstract or complex! Funny, ...I'm in college and I'm learning how to add and subtract all over again. :-)

3.1, due on January 16

Difficult: There are lots of definitions to memorize in this section, but they are all pretty straight forward. I just don't get the "1R not equal to 0R" in the definition for an integral domain and a field. So {0R} isn't an integal domain or a field. Is that all it means?

Reflective: So now that we have a definition for a ring, so what? What does a ring do for us? Now I know why we learned about primes in modular arithmetic. They're important. For example, the integers mod p when p is prime is a field.

2.3, due on January 14

Difficult: Every proof in this section can be followed step by step. But when I finish following the proof, it's hard to get an overall idea of why is works. So can we go through the proof for Thm 2.8 in class and maybe the proofs for Corollaries 2.9 and 2.10? I'd think I just need to go through it again.

Reflective: So I guess that the equations in number 10 from 2.2 were a lot more important than I thought they were. Hmmm. And how do these relate to rings and groups? Guess I'll find out.

2.2, due on January 12

Difficult: This section was a little easier because modular addition and multiplication are so similar to standard addition and multiplication. But one question for verification: Can you only perform modular arithmetic with congruence classes in the same mod n?

Reflective: It's nice that the first 10 properties of standard addition and multiplication apply to modular addition and multiplication and it's kind of cool that 3 x 4 = 0 in some cases.

2.1, due on January 9

Difficult: Section 2.1 is difficult because, although it makes sense, it is difficult to remember because it isn't something I've used a lot. Divisibility and factoring from the previous sections are part of regular arithmetic that I've used my whole life. It's second nature to me. But modular arithmetic is something I've only used on occasion. Since I don't remember it well, it's going to be hard to use in proofs.

Reflective: Modular arithmetic is interesting because I've actually used it in Computer Science 124 (java programming) a few times. So I know it has practical application. Also it's interesting to see how it matches up with remainders. I wonder, too, how it applies to rings, groups, or both.

1.1-1.3, due on January 7

Difficult: The most difficult part of these sections is using the Euclidean Algorithm to find a linear combination of two large numbers for a gcd. Small numbers aren't a problem, but large numbers are. The Division Algorithm and Thm 1.10 (all numbers can be written as a product of primes) were longer and I couldn't reprove them yet, but they make sense and I have seen them before.

Reflective: I found divisibility to be the most interesting part of these sections because the proofs in the book and homework were straightforward, fun little puzzles. I know 371 is all about groups and rings, even though I don't know what those are yet. I wonder how important primes are to them?

Introduction, due on January 7

1.
I am a sophomore and a math/math ed major.

I have taken Linear Algebra (343), Differential Equations (334), and Theory of Analysis (315).

I am taking this class for fulfill my requirements for my major and I might specialize in theory because I enjoyed 315 (?).

Dr. Lawlor (taught me 315) was an effective teacher because he is very organized. He gave a list of concepts we needed to know at the beginning of each chapter so we had lots of time to study and knew exactly what he expected. Also he explained abstract concepts well by using metaphors and lots of examples and he did a great job making himself available to help students during office hours and one on one.

I'm from Mesa, AZ and I LOVE the heat.