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.