4.3, due on October 3

1. The most difficult thing for me from this section was following the proof of Theorem 4.11.  Even that wasn't very hard; it just took a careful reading.  Most of the material in this chapter is familiar.

2. It's interesting to see how most of the theorems are similar to theorems about primes in Z.  I hadn't really thought about common properties between prime numbers and irreducible polynomials before.

4.2, due on September 30

1. The hardest part for me is probably keeping track of which polynomials are which in the proofs. That's mostly because the notation for polynomials is a bit more complicated in general than for numbers.

2.  I thought it was interesting that there is always only one gcd for any two polynomials, even though the gcd is simply the common divisor with the highest degree.  It's not immediately obvious that there is only one monic polynomial that satisfies this definition, but it ends up being true.

4.1, due on September 28

1. The most difficult thing for me is thinking of x as a specific element of the ring rather than a variable.  It makes sense logically, but I'm just not used to it yet.  The example with pi does help with that.

2. I thought it was really cool how well the division algorithm in F[x] correlates with the division algorithm in Z.  Both are fairly simple theorems that are extremely useful later.  I haven't read ahead in the book, but I've heard hints of some interesting ideas that are coming - cosets of functions modulo a polynomial, etc.  I'm looking forward to it.

Test preparation, due on September 26

I think the most important theorems out of the ones we've studied are:
-Every field is an integral domain (and the definitions that say that both of those are commutative and have identity)
-The division algorithm
-Properties of rings that don't come from the definition
-Properties of modular arithmetic

I expect to see questions on the exam that are fairly similar to the homework - proofs that are fairly straightforward but require an understanding of the concepts.

Before the exam, I would like to better understand the proof that every finite integral domain is a field (and the related proof that cancellation is valid in an integral domain).  I will need to review those before taking the test.  I would also like to understand quotient fields of integral domains better, but that will probably come as I do the homework due Monday.  (I won't be in class on Monday, but these are things I need to study on my own.)

Class feedback, due on September 21

I usually spend about an hour on each homework assignment.  Some of the easier ones have taken less time than that, but I haven't spent much more than an hour on any of them.  Proofs in general tend to come naturally to me, and that is what most of the homework has been.  Also, I've seen most of the concepts in the first two chapters before.

Probably the thing that has contributed most to my learning in this class has been doing the readings before coming to class.  That way, when I hear a concept in lecture it's familiar to me, and if there is a proof I didn't understand I can pay much closer attention to it to understand it better.

3.3, due on September 19

1. I had a harder time understanding a homomorphism than an isomorphism.  It still feels like just an abstract property to me rather than a description of the function.  Isomorphism makes much more sense; it's a way to tell if two rings are essentially the same.

2. We have been learning about isomorphic vector spaces in one of the ACME classes (Math 344), so the concept of an isomorphism is vary familiar.  For rings, however, there are many more nice properties than we covered for vector spaces, such as the fact that isomorphisms preserve commutativity and units.

3.2, due on September 16

1. The term "unit" was a bit confusing to me at first.  Normally when I've seen it before it means something that is equal to 1 in some way, such as the unit circle or a unit vector, but here it means something that has a multiplicative inverse (so that, for example, every nonzero element of R is a unit).

2.  I thought it was really cool that every finite integral domain is a field.  That's a result I never would have expected, and it's a bit mind-boggling to think that Axiom 11 implies Axiom 12 in the finite case but not the infinite one.

3.1 part 2, due on September 14

1. The most difficult part was probably keeping the three plus signs straight in the Cartesian product example.  The example itself wasn't that difficult, though; it makes sense that the Cartesian product of two rings is itself a ring.

2. The conditions required for a subring reminded me of the conditions required to show that a subset of a vector space is a subspace (from Math 313 and my ACME classes).  In both cases, a harder problem becomes easier when it's inside something that already satisfies the conditions.

3.1 part 1, due on September 12

1.  One thing that was difficult for me was understanding integral domains.  Are there any fairly simple examples of integral domains that aren't fields?  There were good examples of everything else in the textbook (noncommutative ring with identity, commutative ring without identity, noncommutative ring without identity, and field), but not of integral domains.

(After I wrote this, I realized that Z itself is an integral domain that's not a field.  Silly me.)

2. It was really interesting to try to wrap my brain around the example on pages 43-44.  It's the kind of example that you wouldn't expect to have any nice properties just from looking at the operations, but it does.

2.3, due on September 9

1. The most difficult part for me was understanding what Corollary 2.10 was saying.  I eventually got it, but I had to think about it for a little while, especially to understand why (a,n) = 1 is required but (b,n) = 1 is not.  It makes sense now, though.

2. It was interesting to see all the nice properties of the integers modulo a prime.  I thought it was really cool that every number except 0 has a multiplicative inverse, turning it into a field.

2.2, due on September 7

1.  Probably the most difficult part for me was the "new notation" section, especially distinguishing between exponents in the normal integers and numbers in the Z_n ring.  Even that wasn't very hard, though; I've seen all of these things before.

2.  I really liked the non-example they gave to show why we had to check our definitions of addition and multiplication for modular arithmetic.  It goes to show that we really do need to prove everything we do in mathematics.

2.1, due on September 2

1. The most difficult part to understand was probably the proof of Corollary 2.5.  I had always just assumed that all the equivalence classes were unique, but I guess that needed to be proved just like everything else we learn in this class.  Even that proof wasn't too hard; this section's material doesn't seem to be very difficult.

2.  This section's material is mostly things I have learned before, in Math 290 and in other books.  I'm looking forward to learning about other examples of rings.