6.2, due on 2/28

1.  The concept of the kernel is very familiar but can it also be called the nullspace?  Theorem 6.13 was pretty heavy but I think the post-theorem paragraph made it clear.

2.  Theorem 6.12 was very logical and the proof was a little more complicated/long than I thought it would have to be.  The proof of 6.13 made sense but did not give much insight into the importance of the theorem.

6.2, due on 2/25

1.  The book is very good at building on stuff it has already done but I feel like it falls short in explaining how these concepts are more abstract.  I think it makes sense but it is not exactly clear to me why the quotient ring is any different.  It feels we gave a new name and notation to something that already had a name and coefficient.

2.  The explanation of theorem 6.8 and definition of addition and multiplication of cosets was very straightforward.  Theorem 6.9 seemed trivial and the definition of quotient ring seemed redundant.

6.1, due on 2/23

1.  The definition of coset was unclear to me but the examples helped somewhat but not completely.

2. Theorem 6.4 seemed very logical and the proof very straightforward.  The same thing with theorem 6.5.

6.1, due on 2/22

1. The ideas in abstract algebra are getting more abstract, go figure.  I feel like it has been developed well so it is not that hard to follow.  I did not follow the example after principal ideal generated by c part.

2.  The idea of an ideal is interesting.  The examples depicted the point very well.  Theorem 6.1 was very straightforward and the proof was practically trivial.  Theorem 6.2 also seems pretty trivial/obvious.

5.3, due on 2/18

1.
All the proof were straightforward and/or cop-outs from previous chapters.

2.
In the first example they kind of used the trick we learned in class but they didn't explain it very well, tsk tsk.  It is interesting that F[x]/(p(x)) contains a root of p(x). The discussion of how R[x]/(x^2+1) is isomorphic to (or a definition of ) the complex numbers was very interesting.

5.2, due on 2/16

1. Theorem 5.7 seems strange to me.  I followed the proof I guess I just don't understand the importance of it.

2. Theorem 5.8 was a cop out.  It is the same thing as 5.7.  Theorem 5.9 was pretty interesting.  This section was pretty weak.  It was short and not very interesting.  At least it was easy to follow.

5.1, due on 2/14

1. This section seemed almost completely trivial to me.  Cor 5.5 took a couple seconds to decipher what it was trying to say but I got it.

2.  So we're combining chapter 2 with chapter 4.  This made all the proofs pretty much trivial.  Cor. 5.5 was slightly interesting.  The set of congruence classes of modulo p(x), degree n, is a subset of S, the set of polynomials degree less than n.  I think they worded it strangely.  I like my explanation better.

9.4, due on 2/11

1.  I thought it was strange how we jumped five chapters and only for one chapter.  I'm not sure why they call it the field of quotients when they could just call it the field of rationals but whatever.  I'm not sure that I fully understand Theorem 9.31 or its importance.

2.  I feel like Theorem 9.31 is very important but I'm not quite getting it.  I also feel like we went from being simple to abstract back to simple again.

Test Reflection, due on 2/9

• Which topics and theorems do you think are the most important out of those we have studied?
• The ones with names are almost always the most important, so the well-ordering axiom division algorithm, the Euclidean algorithm, the prime number theorem, the remainder theorem, and the factor theorem.
• What kinds of questions do you expect to see on the exam?
• I plan to see the questions like the homework with at least one proof of a theorem from the book, such as the remainder and factor theorems.
• What do you need to work on understanding better before the exam?
• I just need to review it all, especially a couple of the tricky computational problems from towards the beginning of the semester.

4.3, due on 2/4

1.  I'm not sure if I understood the definition of a unit before.  It seemed trivial to me but Theorem 4.8 seemed very clear and non-trivial to me.

2. I thought it was interesting that in fields irreducible is prime and reducible is composite.  I also thought it was interesting that every polynomial of degree 1 is irreducible, which means that in a field there are an infinite number of primes (irreducibles).

4.2, due on 2/2

1. It wasn't completely clear whether or not monic meant only the leading coefficient is 1 on just the highest degree polynomial or all of the non constant ones.  Also is 5 monic or is 1 monic?

2.  I thought it was interesting how most of the proofs were one-liners because they were the same as chapter 1 proofs.  I also thought it was interesting how again the simplistic, practically trivial stuff from chapter 1 applies directly to these abstract fields.