9.4 Due February 27

The hardest thing I found in the reading today was the last theorem of the section, theorem 9.31. I had to read it a couple of times before I understood what it was saying. And I still don't exactly why theorem matters. The book didn't seem too excited about it and din't offer any examples or anything.

The thing I reflected on most was how to use this same rigorous process to define fields of quotients on other integral domains besides Z. Like polynomial rings, congruence class rings and some of the more abstract quotient rings.

Due February 25

I think the most important things we've studied since the last test have been ideals and quotient rings and how quotient rings are the generalization of the other rings we've studied.

I also think that this last chapter we've done has been the most difficult and thus, is the one I need to study the most.

I would like to see a problem similar to #40 of 6.1 done in class, because I had a lot of trouble with that one.

6.3 Due February 23

This whole chapter has been pretty out there and I think this section has been the hardest to wrap my head around, but I really liked it just the same. The hardest part to understand would be the definition and why that makes a R/M a field. But looking at and thinking about the examples helped me more than anything.

In thinking about the section, I just wonder what kinds of ideals would be prime in some of the rings we've been looking at. Like the ring of continuous functions.

6.2 Due February 20

Well, I didn't have too much trouble in understanding the material. I guess it took me a little bit to realize how the natural homorphism pi is really just forming what is analogous to the rings of congruence classes we've been looking at with Zn and F[x]/p(x).

So I guess my extrapolation question would be what quotient rings would also be field, since some in Zn and F[x]/p(x) were and some weren't.

5.2 Due February 11

The only difficulty I had with this reading is seeing the importance of some of the latter theorems concerning whether F is a subring of F[x]/(p(x)), because my initial understanding of the concepts renders the theorem obvious since the constant polynomials in F[x] will still be distinct congruence classes and if you were only to look at their multiplication addition tables, you would have the same thing as F. It'd be nice to know if I'm missing something here...

In reflecting on the material, it appears that we can construct F[x]/(p(x)) to be a field itself provided p(x) is reducible, because then the gcd of p(x) and all the members of congruence classes will be 1.

5.1 Due February 9

The reading was fairly easy, because almost all the theorems were just previous congruence class arithmetic theorems applied to polynomial congruence class arithmetic. I guess the hardest part was really understanding corollary 5.5 and its implications.

In reflecting on the subject material, I just think about how complicated some of this arithmetic could potentially become. Especially if you consider rings of the form Z_n[x]/(p(x)) and have to kinds of modular arithmetic going on. Or what if you used that kind of ring to form another polynomial ring to make something like (Z_n[x]/(p(x)))[x]...

4.4 Due February 4

The only difficulty i had was being convinced of the remainder theorem. However, after reading the proof and trying out some out examples, I was convinced.

The only reflection on the reading that I had is that polynomial functions is really the same thing that we've been doing since middle school, only with a more formal approach. I'm curious as to what benefits one can gain from looking at the concept through these formal lens.

4.2 Due February 2

The thing that was most difficult for me was the part about if f(x)|g(x) then cf(x)|g(x). I find this weird because that's definitely not like normal arithmetic. The other thing that bugged me was why the gcd of two polynomials is the highest degree monic polynomial that divides both polynomials. This bugged me a little because of all the potential cf(x)'s that could be greater common divisors. However it makes sense, because we could have an infinite number of divisors and really we just need one to be defined as gcd which can be defined in all cases and the monic polynomial fits that role.

The thing that this reading made me think about is that if we can have relatively prime polynomials, can we have prime polynomials? How about unique prime factorization of polynomials? This is interesting stuff.