Section 7.1 (part 2), Due Wednesday, February 28

The section for today was the second half of 7.1.

I thought the most difficult thing about this section was rotation groups.

I like that rings can be made into groups. I thought that it was interesting that every ring was an abelian group under addition.

Section 7.1, Due Monday, February 27

The section for today was 7.1: Definition and Examples of Groups.

The most difficult thing in this section was finding inverse permutation funtions to get the identity.

Judging on what I have read so far, groups seems like they will be interesting to learn about. I am curious to see how similar/different they are compared to rings.

Section 6.3, Due Friday, February 24

The section for today was 6.3: The Structure of R/I When I is Prime or Maximal.

The most difficult thing about this section was idea of maximal rings.

I would just like to reflect on rings in general seeing as this is the last section about rings. I think that they are very interesting and have a lot of convenient properties. It's cool that the same properties hold for the very general cases.

Section 6.2 (Part 2), Due Wednesday, February 22

The section for today was the second part of 6.2: Quotient Rings and Homomorphisms.

The most confusing/difficult part of this section was the proof of theorem 6.13, it was hard to follow.

I thought that this section was interesting. I am wondering how this use of the term kernal relates to the kernals I learned about in Linear Algebra and Multivariable Calculus. I guess they seem similar.

Section 6.1 (part2) and section 6.2 (part 1). Due Tuesday, February 21

The reading for today was the rest of section 6.1: Ideals and Congruence, and then the first half of section 6.2: Quotient Rings and Homomorphisms.

The most difficult concept for me was the idea of "cosets."

I am wondering how general these ideas can possibly get. It seems that each section just generalizes further than the last, making it more "abstract." I guess that is the point. I believe that we are near the end though. The next chapter is about groups, but I am sure we will be using some of the ideas from the first six chapters.

Section 6.1 (part 1), Due Friday, February 17

The section for today was the first half of 6.1: Ideals and Congruence.

I thought that the most difficult thing to understand about ideals is the distinction between principal ideals and non-principal ideals.

I thought that the concept of ideals was very intriguing. The absorption property was very interesting--a ring that behaves ideally.

Section 5.3, Due Wednesday, February 15

The section for today was 5.3: The Structure of F[x]/(p(x)) When p(x) is irreducible.

The hardest part to understand was theorem 5.11, the fact that the extension field contains a root to p(x).

I thought that it was fascinating that the definition for the complex numbers comes from these ideas and structures.

Section 5.2, Due Monday, February 13

The section for today was 5.2: Congruence-Class Arithmetic.

The most difficult part of this section was Theorem 5.9. I don't see how a non-constant polynomial can be a unit.

I think that it is great that addition and multiplication, along with other theorems transfer to F[x] so nicely.

Section 5.1, Due Friday, February 10

The section for today was 5.1: Congruence in F[x] and Congruence-Class Arithmetic.

The most difficult part of this section, for me, was the concept of infinitely many congruence classes for R[x]/(x^2+1).

I think that it is really interesting/convenient that the ideas and rules about congruence classes transfer so nicely to the polynomial rings.

Sections 4.5 & 4.6, Due Wednesday, Feb 8

The sections for today were 4.5: Irreducibility in Q[x] and 4.6: Irreducibility in R[x] and C[x].

The most difficult thing in these two sections, for me, was Einstein's Criterion (Theorem 4.23) and its proof.

I was very interested to learn the proof of the Rational Root Theorem. I have used it a lot in various math classes and while tutoring in the math lab. It is nice to learn where it comes from.

Questions about Reviewing for the test:

Which topics and theorems do you think are most important out of those we have studied? The Division Algorithm, Rings and their properties, isomorphisms and homomorphisms

What kinds of questions do you expect to see on the exam? I expect to see problems asking if sets are rings, if two rings are isomorphic or homomorphic to eachother, proofs about divisiblity, factoring and dividing polynomials, functions induced by polynomials, etc.

What do you need to work on understanding better before the exam? I need to read through my notes and study the proofs listed on the review sheet. I also should go over the properties of rings, fields, and integral domains.

I would like to see this problem worked out: If there exists a ring isomorphism A

B we write A = B. Prove that = is an equivalence relation

on the class of all rings. .

Section 4.4, Due Monday, February 6

The section for today was 4.4: Polynomial Functions, Roots and Reducibility.

The most difficult thing about this section was the Proof by Induction of Corrollary 4.16.

I thought it was very interesting to learn about the distinction between a polynomial as an indeterminate or transcendental element and as a function induced by a polynomial element.

Section 4.3, due Friday, February 3

The section for today was 4.3: Irreducibles and Unique factorization.

I think that the most difficult part of this section is the idea of "associates."

Again, I think that it is very interesting that these properties of irreducibles (primes) carry over from the integers to polynomials.