6.3, Due February 29th

1)  For the definition of a prime ideal, I thought that whenever bc was in P (the ideal) that b is in P or c is in P.  I thought that was just a part of the absorption property of ideals?  I guess it just means that it is possible for there to be a product of two things that are not originally in P, but the product is in P?  Also, I don't fully understand why an ideal containing the 1 element is actually the entire ring.  Is there any way we could go over that a little bit in class?
2)  Maximals are a super interesting idea to me.  But how could you ever really know that you couldn't get bigger than the ideal unless you had the entire ring?

6.1 and 6.2, Due February 24th, 2016

1) In the example at the bottom of pg. 140, there is an example that talks about the cosets of I.  I'm so sorry but what does it mean to be a coset of I?  It also talks about the multiplication of cosets on pg. 146, and I just wish I understood better what exactly a coset is.
2)  It is interesting now to see these al relate to homomorphisms that we talked about before, and also how they differ.

6.1, Due February 22

1) The definition of an ideal on pg. 135 (second edition) is interesting - maybe it is not a commutative ring, but it is still possible for r*a to be in I as well as a*r.  I guess what makes an ideal different than a subring?  Does an ideal need to be a subring?  If something is countable is it considered finite?
2)  I would be interested in understanding better the usefulness of ideals; why are we interested in them?

5.3, Due February 19th

1) The idea of F[x]/p(x) having a root of p(x) is still a little strange to me if we can focus on that in class.  This is also probably silly but the way that we can know if something is irreducible is if it has no roots?  Even the extension field K that contains the root doesn't make too much sense to me.
2)  It is interesting to see that although the rest of fields modulo polynomials has pretty much mirrored the moduli that we studied before, but now for p(x) to be irreducible, there are a lot more characteristics.

5.2, February 17th 2016

1)  This might be a silly, but I don't know what it means when in the example after the proof of 5.7, when it says "we now have a ring that has Z2 as a subset."  What exactly does this mean and why is it surprising?  So surprising, that they say later, "However E does contain Z2 as an honest-to-goodness subset, without any identification."  Also, what does it mean to say without any identification?
2) Arithmetic is more interesting in this congruence-class arithmetic.  However, it still is very similar to the arithmetic that we learned in Zn.

5.1, Due February 16, 2016

1)  I like this section.  What are differences between polynomial congruence classes and just normal congruence classes in Z mod n?  The most confusing part for me was probably the examples at the end of pg. 122.
2)  I just like that all of the proofs are pretty much the same as proofs for congruence classes Z mod n.  Is the proof at the bottom of pg. 121 in the second edition rigorous?  I guess what I mean is it just seems like a general explanation.

4.5, 4.6 February 12

1) I did not really understand the proof of Eisentstein's criterion and why that woul dbe true... Is there a time when p^2 would divide the leading coefficient?  Why are the so many less irreducible in R(x) and C(x), especially when we are dealing with irrational numbers?
2) It is cool to me to think that every polynomial in R(X) is reducible, except for first degree polynomials.

4.4, Due February 10, 2016

1) For the factor theorem on pg. 102, what if you deal with something like bx-a.  Is there a theorem that tells us that a/b is a root of the polynomial?  Also, how do we know when we need to use induction for a proof (like on the proof of corollary 4.16).  How can you prove that something does not have roots?
2) I am just curious when we can switch fields to get roots, like how sometimes we say a polynomial has imaginary roots?

4.3, 2/8/16

1) So can you basically consider an associate to be something that is divisible by a certain polynomial?  And according to my understanding, it also means that if a=bu, then b=au^-1.  In the definition for irreducible polynomials, is it suggesting that some polynomials are divisible by anything?  I guess I can't really see the difference between "associates" and just being divisible by something else or reducible.
2)  Do irreducible polynomials sort of act like primes in certain ways?  Or do we sort of consider "associates" to be like a prime factorization of a polynomial?

4.2, Due February 4th 2016

1) I guess I don't really understand how to find the GCD of polynomials.  Also what exactly is a monic common divisor?
2)  It's nice to get good at all of this polynomial division and factorization for various integration problems I face in the math lab! haha

4.1, Due February 3, 2016

1)  I guess they just didn't go through a ton of the proofs of theorem 4.1 and I'm just wondering if we are going to go through those in class?  I think for the most part proofs involving polynomials aren't very difficult, per say, but more tedious.  But because there are so many variables and constants going on, I do get a little bit confused.  Also, why does it help us to know what the degree of the polynomial is?
2) I thought it was interesting that you could take any ring R (even if it is not a commutative ring) and there exists a ring P that looks like it does have commutativity, looking at the second part of theorem 4.1.