Section 4.4 due 9/29

Is the difference between f(x) and F[x] that f(x) is the function of F[x], meaning that F[x] does nothing? When the corollaries are talking about a function having no roots, do they mean that they don't have real roots (but there exist complex roots)?

The most interesting part of the section was Theorem 4.14. At first I thought it meant you didn't have to do polynomial division--you just had to subtract f(a) from the polynomial--but then I realized that it just meant that it gives the remainder. Despite the disappointment, I always love how math works out perfectly.

Sections 4.2-4.3 due 9/27

If we had common divisors of f(x) and g(x) that were monic and had the same degree, how do we tell which one is greater? Is it the one with the greater coefficients? (For example, what is the gcd if the two divisors are (x+1) and (x-1)? Or would it be the product of the two if those were the divisors?)

Most of these theorems are the same as the theorems for the integers. The ones that are specifically geared toward polynomials are about ir/reducibles and the degrees of reducibles.

Section 4.1 due 9/24

I understood Theorem 4.1 until part iv--is it saying that the bi = 0R because the coefficients are equal to 0? I don't really understand part iv.

I like the familiarity of this section. I started learning about polynomials years ago (middle school or high school?). And the rules are still the same. I like dividing polynomials. Although this class is all about abstraction and proving things, which is not what I did with polynomials in high school, I think it is the start of what will be an interesting chapter.

Exam prep due 9/22

Some of the topics that we have studied that are of utmost importance are greatest common divisors, the Fundamental Theorem of Arithmetic, congruence modulo n, and the definitions of rings, identity, fields, etc.

I expect to see a lot of definitions that require specific details for full credit (after all, those details are crucial), proofs about the above topics, and proofs that we have done in class. I expect that it will be a fair grade of how much I have prepared for it, which will (hopefully) be a good thing!

Section 3.2 due 9/17

I don't understand the discussion on the zero divisor (or why it is important, rather) because it seems like only a unit would be important. Also, are there zero divisors in rings other than matrices and modular arithmetic?

I really like this section. You can never assume that all functions in a ring are valid--it has to be defined. BUT you can use previously shown axioms to show that things like subtraction work, which is awesome! Everything works out perfectly, which is what we, as mathematicians, want to happen.


Warning: Math joke below.

How does one insult a mathematician?

You say: "Your brain is smaller than any > 0!"

Section 3.1 (con't) due Sept 15

• I have spent about 3-4 hours on each homework assignment. The book is helpful, the lectures are more helpful (because they clarify what the theorems are saying), but they do not feel like preparation for the homework. Ultimately, I think I just need to keep at it to improve my creativity for the proofs. A lot of times, I get stuck on the proofs because I don't know what is sufficient or what the next step is.
• I think the things that would help me learn more effectively are things that are up to me: stop by your office hours and study with a group.
  

Section 3.1 due Sept 13

I don't understand the definition of an integral domain (where it says 1 sub r is not equal to 0 sub r and the discussion of the zero ring, in particular).

I like the idea of a ring--it is a subset that has solid properties. I like that the set of integers mod n is a ring. It seems like the concept of rings will be really useful in application and in proving things.

Section 2.3 due Sept 10

I don't really understand Theorem 2.8 and what the set of all congruence classes modulo n, with n being a prime number, has to do with ax=1.

The cool thing about this section is that some of the formulas look similar to those that I saw in Linear Algebra, such as Theorem 2.11. The difference is that those dealt with matrices but these equations deal with congruence classes.

Section 2.2 due Sept 8

The most confusing part of 2.2 was p 33 with the addition and multiplication tables.

To me, the properties of the classes were interesting because they are the same properties that we learned in Linear Algebra that must hold true for subspaces. As those are subspaces within a space, these are subsets within the integers.

Section 2.1 due on Sept 3

The most difficult part of the material was the concept of congruence classes. Theorem2.3 describes equal congruence classes, but aren't there only supposed to be n classes?

From what I do understand of congruence classes, it seems like a cool idea. It seems like it relates to the form that the division algorithm takes in because an integer is multiplied by something and then a remainder is added to get the desired integer.