6.3, Due March 2

Difficult:

I don't understand their example at the top of pg 155. They say that the principle ideal (x) in Z[x] is prime, but since the quotient ring Z[x]/(x) is isomorphic to Z, then Z[x]/(x) is an integral domain but not a field. How does that isomorphism make it so that it can't be a field?

Reflective:

I think the definition (and explanation) of a prime ideal was the only thing that made complete sense in this chapter. It made logical sense to me to go from saying that when p|bc then p|b or p|c to saying when bc is in (p) then either b is in (p) or c is in (p).

6.2 part 2, Due February 28

Difficult:

I had a hard time understanding homo/isomorphisms previously... I can't imagine this is going to be any different. Now we're adding in the idea of a kernel, which I also am confused about. The kernel is basically just a zero map from what I understand of the chapter, but the way they sometimes talk about it makes me wonder if that's really what it is.

Reflective:

I think I'm in luck today! It usually happens that I do my blog 2 days before it's due, before class. As in, today we're talking about the FIRST part of 6.2 and I'm already doing the blog for the last half. Sometimes it can mess me up a bit, but today I'm in luck. Question #25 from homework 6.1 is proving thm 6.10! Now I can check my work and fix any mistakes :)

6.2 part 1, Due February 25

Difficult:

I think what will be the hardest to work with is the notation of addition of cosets. Just looking at one of the simpler examples of an addition and multiplication table on page 146 got me worried about what I would need to do with larger fields...

Reflective:

The addition and multiplication seems to be similar to that in Zn, but now we have to add a "+I" at the end. As long as I'm correct about that, it shouldn't be too hard to do that for all the new multiplication and addition that I need to do...

6.1 Part 2, Due February 23

Difficult:

At the top of page 140 when the describe the congruence class of a modulo I, I got confused. Their first set up with b-a in I makes sense. I don't understand why they carried on further from there... Why was it important to get to a+i such that i is in I? Why did we even need to develop that i in the first place?

Reflective:

This is basically like the congruence in Z, except that they added the idea of cosets and that instead of using [a] to denote the congruence class of a, they say a+I instead.

6.1 part 1, Due February 22

Difficult:

The concepts really aren't too difficult to understand, I'm just concerned that the application will be more difficult than it seems. Generally (and this section is no exception), I have to break down each theorem into it's individual parts and think "ok, so if a subring I of R is an ideal, then there's an element of R and an element of I, and when I multiply them together the product is in I. Ok, that makes sense."

Reflective:

I thought it was really cool when they talked about how the congruence modulo 3 can be characterized as a-b being a multiple of 3. It's really the same thing we've said all along, but either they haven't phrased it that way, or I missed it when they did. That definition makes a lot more sense in my mind than saying that 3 divides a-b.

5.3, Due February 18

Difficult:

I had a little bit of a hard time understanding the idea of the extension field K. The proof of 5.12 wasn't too hard to understand, but I had to break it down a little bit farther and think about how if p(x) is an irreducible factor of f(x) and p(x) has roots, then any roots it has are also roots of f(x).

Reflective:

Well I feel like this is what I've been saying this whole chapter, but this is a lot like the stuff we did back in chapter 2. What's cool though is that while F[x]/p(x) is a field just like how Zp is a field, F[x]/(p(x)) also contains a root of p(x), which is obviously not possible in Zp.

5.2, Due February 16

Difficult:

In the proof of Theorem 5.7, they make a set F* of the constant polynomials and verify that it is a subring of F[x]/(p(x)). What I don't understand is why don't they just state it that way in the theorem? Why don't they just say that the set of constant polynomials make a set F* which is a subring of F[x]/(p(x))?

Reflective:

Again, many of the proofs are just altered versions of the proofs for rings Zn, so the concept presented are easily recognizable.

5.1, Due February 14

Difficult:

Oh boy. And I thought regular modular arithmetic in Zn was rough. I still find the first chapter of that confusing, when they talk about things like 17=5(mod6). It's really not any different here in F[x] modulos except that we're talking about functions now.

Reflective:

It's kind of fun to see them give a theorem using functions and for the proof, just ask you to adapt the proof from chapter 2, using functions in place of integers.

9.4, Due February 11

Difficult:

I was a little confused in the proof of Lemma 9.26 with all the multiplication that was going on. They didn't explain WHY they were doing anything, so it was harder for me to make the logical connections and understand the logic behind the proof.

Reflective:

I mostly just thought it was cool to see these equivalence relations. I hadn't ever really thought about why 1/2 =2/4 and so on and so forth. This is the kind of thing students in junior high and high school get confused about, and as a math ed major, it's cool to see an explanation.

Questions, Due February 9

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

Obviously any theorem with a name, such as "The Remainder Theorem", "The Division Algorithm" and so on and so forth. It seems like these first 4 chapters have been about division, factors, addition/multiplication operations

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

I'd expect some questions about definitions (such as, "What is a ring?"), questions that apply definitions ("give an example of a ring homomorphism"), and questions about theorems (stating, applying, and proving).

What do you need to work on understanding better before the exam?

Mainly I need to work on understanding chapter 4, especially getting a better understanding of irreducible polynomials and how to find them. I also have a hard time with "surjective", more specifically with how to prove that something is surjective.

4.4, Due February 7

Difficult:

I am still struggling with the idea of irreducibility. The homework due today asked us to list some irreducible polynomials. The answers were in the back of the book, but I still didn't understand exactly why it was irreducible. The Theorem 4.18 helps me to understand a little bit better of how to determine which polynomials are irreducible.

Reflective:

I find the remainder theorem to be really cool :) I hadn't realized that if we take the divisor of the form (x-a) then the remainder will be f(a). I'm wondering now, can we always get a divisor of that form? What happens when we don't?

4.3, Due February 4, 2011

Difficult:

I found the concept of an "associate" very hard to understand. The book didn't really provide a good example that could solidify in my mind what it means. If I understand correctly, if a=3 and b=1/2 then u would be 6 so that a=bu? It's much more difficult to think about in terms of polynomials....

Reflective:

The idea of a polynomial being "irreducible" is interesting. It's technically a way of finding "prime" polynomials, but for some reason they thought "irreducible" would be a better term :)