Test review, due October 26, 2016

The most important theorem we've learned is probably the First Isomorphism Theorem (since you said so in class :) ).  In general, I think ideals are the most important thing we've studied, since they include almost all of the other things we've learned.

I expect to see problems on the test that are fairly similar to the ones we've done on the homework, but not identical.  I like being asked to come up with something new rather than redoing problems I've already done.

I could use some more review of prime and maximal ideals, and some more work with ideals in general.  This is both more recent and more difficult than the other things we've done in class.

6.3, due on October 24

1. The idea of a maximal ideal was a bit tricky for me, since we've never covered anything similar before in this class.  It was easier to see how prime ideals connect to the ideas of primes and irreducibles.

2. It was interesting to me that I being prime no longer implies that F[x]/I is a field.  That's something that our intuition from previous chapters definitely wouldn't have predicted.

6.2, due on October 21

1. The most difficult part of this section for me was understanding the functions associated with the proof and examples of the First Isomorphism Theorem.  The functions themselves aren't too hard, but they are obscured by a lot of notation, especially in the proof.

2. We learned about the First Isomorphism Theorem and the canonical epimorphism for vector spaces in Math 344 a month ago!  It's amazing how many similar ideas are popping up in both rings and vector spaces.

6.2, due on October 19

1. I know that "nothing" isn't an acceptable answer for this part of the question, but there wasn't really anything difficult for me about this material.  Cosets were difficult for me when I first learned about them in Math 344, but they behave exactly the same way with respect to ideals as they do to vector subspaces, and the notation is the same.

2. It was interesting to see how the concepts and theorems we have been using for principal ideals in Z and F[x] generalize nicely to quotient spaces with arbitrary ideals.  It's always cool to see connections like that between different mathematical ideas.

6.1, due on October 17

1. I don't understand why R must have an identity for Theorems 6.2 and 6.3.  Wouldn't the proof still work if R was a commutative ring without identity?  Also, are there any good examples of ideals that aren't finitely generated?  Never mind, I just realized that the second example on page 6 is an ideal that's not finitely generated.

2. We learned about cosets of vector subspaces in Math 344 (one of the ACME classes), and Dr. Grant mentioned that those behave just like cosets in a ring.  It's really cool to finally be learning about those, especially after having the background to understand them better.

5.3, due on October 14

1. I didn't understand at first why Corollary 5.12 holds for polynomials that are the products of nonlinear irreducible polynomials, but then I read the proof again more carefully and understood it the second time.

2. I thought it was really interesting that for every polynomial p(x) over a field, there is an extension field where p(x) has a root.  It was also really cool to see the definition of the complex numbers in terms of the extension field modulo x^2 + 1.  That was something I had never considered before.

5.1, due on October 10

1. The hardest part for me was remembering that associates aren't essentially identical in congruence classes the way they are in factoring.  If we're dealing with the congruence class [ (7/9)x + 2 ], we can't multiply by 9 to clear the fractions or multiply by 9/7 to make it monic, since those would be different congruence classes.

2.  I thought it was really cool how easily results from Z mod n translate to F[x] / (p(x)).  Only one of the theorems (Corollary 5.5) had to make any changes, and they were very minor.

4.5 and 4.6, due on October 7

1.Eisenstein's Criterion was a bit confusing at first, especially the seemingly arbitrary condition that p^2 does not divide a_0.  It makes sense after reading the proof, though.

2.I've seen most of the results in these sections before, so a lot of our reading was familiar.  I thought Lemma 4.21 was quite interesting, since it's not immediately obvious that you can't have the coefficients that are divisible by p be split up between g(x) and h(x).

4.4, due on October 5

1. The hardest thing for me is probably keeping straight the actual polynomial and the function induced by the polynomial, especially when both use the same symbol.  The paragraph on true and false statements in particular took a little while to wrap my brain around.

2. It was really interesting to see concepts from high school algebra showing up in an abstract algebra class.  Corollary 4.16 is one good example of this; it's a generalization of the same theorem about roots of polynomials in R.