6.2 (part 2), due February 28

As I finished this section, I realized that I was still having a hard time with the notation R/I.  I just don't know what it means in words. For example F[x] means the ring of polynomials with coefficients in F.  I don't know how to think of R/I in sentence form.  I also noticed that whenever you have a lowercase letter r plus an upper case letter R it seems to always be referring to a coset r + R.  I thought that might be a fiarily obvious assumption but I didn't want to teach myself a lie.  Another question I had was does K always denote the kernel or are there times that K could be just a normal ol', run of the mill ring?

6.2 (part 1), due February 25

Call me crazy, but when I hear "quotient" I think division and I don't see how these quotient rings have anything to do with division.  The notation looks like a fraction which is related to division, but I'm trying to ignore that because it's been throwing me off a little bit from the very beginning. So I guess my question is - is there any sort of division involved in quotient rings? The addition and multiplication of ideals is fairly straight forward when you pay close attention to the notation and what each bit means. As  I read through the section I almost instinctively foiled (a+I)(c+I), but then I realized that I^2 wouldn't make sense. You gotta be careful with this stuff.

6.1 (again), due February 23

I accidentally read all of the section for the last blog. So for this blog, I will relfect on the things that I read for yesterday and see if I can make sense of the things that I had a hard time understanding before.  From our lecture today, I realized that an ideal I is in fact a ring and is therefore a set. I think I understand principals and ideals generated by c_1, c_2,...., c_n a little better, altough they are still a little unfamiliar.  As I reviewed the section, I realized that I was a little shaky on the concept of a coset. Hopefully that will come out in lecture tomorrow.

6.1, due February 22

Boy, Dr. Jenkins! You weren't kidding when you said that the material was going to take one more step in the abstract direction. The concept of ideals didn't seem too different from that of an equivalence class but as I kept reading I realized that I was confused as to just waht an ideal was.  Because an ideal was initially compared to a modulo n, I immediately thought an ideal was similar to an equivalence class.  Then I read the theorem that showed that I was a subring.  Well equivalence classes aren't rings.  So, since an ideal is a subring, then it must be a set.  But I cannot shake the similarity to an equivalence class.  Am I just getting my wires crossed?

5.3, due February 18

This section was...complex. No pun intended.  I didn't really understand why if p(x) was irreducible, then it had a root. Or was it saying tht there was a root in the set of residue classes? That part went over my head.  I was able to follow the first part fairly well because I had the example to follow, but by the time we got to the last page I was confused.  I understand how x^2 + 1 has roots in the comlex numbers, but thee way I was interpreting this section, it sounded like the author was saying that it has roots in the real numbers as well.  I don't think I understand how that works.

5.2, due February 16

First of all, can I just point out how awesome it is that the author used the phrases "honest-to-goodness" and "if doing this makes you uncomfortable"?  When I read each of those lines, I laughed right out loud.  As far as the math of this section goes, it really isn't all that hard...until you get to page 127. Once I got to that point, I was lost.  The math started to get more theoretical and less concrete.  I was waiting for an example to explain the theorems but there wasn't one! It kinda left me hanging....

5.1, due February 14

This section was rather straight forward since it was so similar to the sections on modular arithmetic.  It's almost as if they go together like peanut butter and jelly.  On the last page of this section, they gave an example of a polynomial in R[x] that had an infinite number of congruence classes and then they gave an example of a congruence in Z_n[x] that had n^k distinct congruence classes.  This got me to thinking, "Is there only a finite number of congruence classes in Z_n or are there other cases that have this distinciton?"

9.4, due February 11

Ok, so the resounding question from this section was "What?" For some reason I had to read this section twice before any of it clicked.  The first time through things just didn't make sense.  I was switching equalities (especially the addition formula) in my mind so as I read the proofs everything was out of control.  I thought it was really cool (after I could make sense of it) that we used fields to prove the addition and multiplication of fractions.  I had done that in one of my education classes with diagrams and now I can do it arithmetically. Neat!

Exam 1 reflections, due February 9

On this first exam I'm expecting to see a few practical application problems and proofs of major theorems.  I imagine it being structured much like a homework assignment.  I could also see the possibility of a few definitions being on there, but I would expect these to be the less common definitions that don't necessarily apply to many theorems.  The more common definitions will show up in the proofs of the theorems themselves.  I would think that we would be asked to state and prove big theorems like the handful that have actual names (the Division Algorithm, the Fundamental Theorem of Arithmetic, etc).  The one thing that I am the least confident on is associates.  That is one thing that I need to practice and study more just to try and make sense of it before I take the exam.

4.4, due February 7

OK. So apparently everything my algebra teacher told me was a lie.  Well, maybe it wasn't a lie but it wasn't completely accurate.  I always thought that "x" only had one meaning in a function.  Actually it has two and it depends on how you are thinking about it as to which meaning "x" takes on.  If you are thinking of "x" in the context of R[x], then "x" is an element.  If you are thinking of "x" in a polynomial function, "x" takes on the standard algebraic role of the variable to define a function f: R -> R.  I felt like most of this section was clarifying all the falsities our algebra teachers taught us in high school.  The one thing that I have found that frequently gives me grief is distinguishing between f(x) and F[x].  When you read them to yourself or out loud to your imaginary friend, they sound the exact same, "f of x". I often find myself confused about which f, and now, which x I'm talking about.  Is that big F or little f? Is x the variable or is it an element? PotAtoe, potatoe? I have a feeling we can't call this one off....

4.3, due Febraury 4

I think I understand what this section is TRYING to say but I'm having a hard time understanding how it's saying it.  I get the most confused when the proof of a theorem just says to adapt the proof of a related theorem for the integers.  As far as I understand, an irreducible polynomial is analogous to a prime number and a reducible polynomial is analogous to a composite number.  I think I understand the concept of an associate of an element but I would like to see more examples of the application of associates.

4.2, due on February 2

This section was a little mind blowing.  They started by claiming that the divisibility and greatest common factors carry over to polynomials and I thought, "Cool!" But then they showed that any any constant multiple of a function was also a multiple.  That part made sense, but then we hit GCD.  How in the world can you have a GCD if you can just multiply it by a constant and get a bigger one? That's where monic polynomials come in.  A monic polynomial is a polynomial that has a leading coefficient of 1.  The gcd fo a polynomial has to be a monic polynomial.  So if you use the Division Algorithm and get a polynomial that is not monic simply multiply it by the reciprocal of it's leading coefficent and BOOM! you get a monic polynomial.  *mind blown* I'm just going to have to remind myself to remember the monic polynomial.  That's the tricky part.