7.1 part I, due on March 2

Difficult: This section was pretty straight forward. But it's probably going to get abstract really fast. So I should make sure I get the definitions solid right now.

Reflective: I remember doing Permutations and Combinations in Algebra II. I remember that it made sence then, but I don't know remember much about it. I wonder how similar groups are to rings.

9.4, due on February 27

Difficult: If R = integers and F = rationals, then what is Rstar in Lemma 9.29? Because it seems like Rstar = integers. And what is K in Thm 9.31? The real numbers.

Reflective: I like this section! I've never thought of the rationals this way before and its kinda cool and makes sense. This section isn't too abstract.

Review QA, due on February 25

The five theorems listed in class I need to know with their proofs. Also, Cor 6.16(when max ideals are prime), thm 6.11(K=0 iff f is inj.), thm 4.11 (equivalent statements to p(x) is irreducible in a field), and 6.15 (maximal ideals and fields)and all the thms that generalize congruence classes and their properties to other rings and quotient rings other than the integers mod p and fields mod polynomials.

I need to know the definitions: maximal, prime, kernel, natural homomorphism, quotient ring, ideal, principal ideal,cosets, and greatest commone divisors . Also I need to remember that a+I is in I iff a+I = 0+I in R/I. And a-b is in K iff a is congruent to b modK. And I need to know when a polynomial is irreducible. I need to get better at figuring out what cosets "look like".

Questions:
Can we do some examples of what cosets "look like" in tricky quotient rings?
Can we define as a class: quotient rings, cosets, ideals?
Can we cover some examples of prime ideals that are not maximal?

6.3, due on February 23

Difficult: It all makes sense if I take it nice and slow, but I feel like it adds to the ton of information I'm already trying to remember. I'd like to go over the proof for Thm 6.15 again.

Reflective: So let me recap. An ideal P is prime iff R/P is an integral domain, R/P is a field iff P is a maximal ideal and a commutative ring R with identity's maximal ideals are all primes.

6.2 part II, due on February 20

Difficult: I'm not understanding entirely how Thm 6.11 works, about the kernel being 0 iff the homomorphism is injective. The rest I have a vague understanding of.

Reflective: Very abstract, but I think pictures of drawing pictures of the sets' elements mapping to their image helps a little bit.

6.2 part I, due on February 18

Difficult: So I know that quotient rings are analogous to the integers mod something or F[x]/(p(x)). But what is the difference? I am still trying to figure out exactly what it is. What does the book mean when it says: "One sometimes speaks of factoring out the ideal I to obtain the quotient ring R/I."?


Reflective: This is abstract stuff. Hopefully I catch up with it in class.

6.1, due on February 17

Difficult: I'm still a little confused about cosets, but the rest makes sense. So the coset of a+I is where a is the remainder I guess. The last example in the section just doesn't make sense. Weird notation.

Reflective: So a+I is like [a] and modn is like modI where I is a subring and ideal of some R. I think I got it.

5.3, due on February 13

Difficult: This section was very abstract. (kinda like the title of the course suggests it will be...funny.) Theorem 5.11 I don't entirely understand. Why is it important? And the proof doesn't make sense yet.

Reflective: Theorem 5.10 makes sense because it is analogous to the integers mod a prime. But 2 of the 3 proofs I don't really follow. Are groups more complicatied or less complicated than rings?

5.2, due on February 11

Difficult: Why do the theorems and definition exclude constant polynomials? And what is so nice about every field containing an isomorphic subring?

Rwflective: The chapters are getting more and more abstract. This makes it a little tricky.

5.1, due on February 9

Difficult: I still don't get the proof forCor. 5.5 very well and I need to prcatice this stuff. It's a little weird.

Reflective: It's nice that mods and congruence classes are similar with polynomials as they are with integers.

4.5and 4.6, due on February 6

Difficult: All of the thereoms, lemmas, and corollaries made sense intuitively, but it took a lot of brain energy to follow the proofs. I don't have them down solid; just the basic idea.

Reflective: These are interesting, helpful patterns to know. I especially like the Einstein's Criterion and the Rational Root Test.

4.4, due on February 4

Difficult: This chapter seemed to be a bit more abstract than the other chapters. And "induced" threw me for a while. So I guess it means that a polynomial acts as a function and so certain things make sense like setting it equal to 0.

Reflective: It's good to see the Remainder and Factor Theorems written out because I have known them, but I always forget about them.