Section 7.3-7.4 Due May 28

1. I'm having a hard time understanding cyclics. And I can't figure out what an automorphism would be with groups like S_3 and D_4. It started getting hazy when it was talking about the image and e_G and e_H, so if we could go over that a lot in class that'd be great.
2. It's interesting that every single infinite cyclic group or order n is isomorphic to Z_n. It was interesting how similar the properties of proofs for subgroups were to subrings.

Sections 7.1-7.2 Due May 26

1. Does part 2 of theorem 7.5 imply that G is commutative, and hence an abelian group? When it started talking about order of an element, I didn't understand what e was. When it says "if a has a finite order n" is n the number of times you have to multiply a to get e? and therefore is n not unique?
2. Groups seem to follow traditional rules with exponentials, and overall still seem fairly simple. It's interesting that we didn't talk about order in rings, but only in groups. The corollary on the top of page 178 was interesting because I wouldn't have thought that such a correlation between the orders of elements would exist.

Section 9.4 & 7.1 Due May 24

1. The definition of addition where [a,b] + [c,d] = [ad+bc, bd] was very random. Theorem 9.31 made no sense. So with groups can you just define * as whatever you want it to be? Because composition doesn't seem like just a standard operation like multiplication of addition or something. I don't get what the actual group is. Is it the set of the bijective compositions? Or the set of the permutations?
2. It was interesting how on the proof of lemma 9.29 they proved it backwards and went from f(a)+f(b) to f(a+b) instead of the other way around. I haven't seen it like that before. Groups seem similar to rings, but with less operations.

Sections 6.2-6.3 Due May 20

1. The definition of prime ideals was a little strange. I didn't understand how there could be infinitely many maximal ideals in Z, nor why the maximums have to be prime. I couldn't remember what a kernal was so all of the maps to the kernal K didn't make sense.
2. I like the idea of ideals and how the new notation is easier. I still have a hard time seeing applications of any of this though. It said something in the book about the Chinese Remainder Theorem, which sounds like something interesting.

Section 6.1 Due May 19

1. I wasn't really sure what they meant by "absorbs". Theorem 6.6 was a little confusing. Are there infinitely many ideals? Finitely generated was also confusing.
2. So according to theorem 6.1, a subset can be an ideal without necessarily being a subring? Ideals seem to make modular arithmetic a little simpler. It'll be interesting to see how these will add to what we have been doing, since they seem to just be a replacement of our current notation of classes.

Sections 5.1-5.3 Due May 14

1.  The extensions didn't make any sense. Especially after they replaced x^2 with x+1. The whole extension field idea was confusing.
2. Everything else seemed exactly like normal equivalence classes. The proofs were the same. The multiplication and addition tables worked the same. I think the idea of equivalence classes with something like polynomials is a little bit silly though. It seemed like it was just taking a simple enough idea and making it as complicated as possible by applying it to polynomials rather than just integers. However I would be interested to here how something like this would be useful.

Sections 4.3-4.4 Due May 14

1. It will be hard to get to this new way of thinking about roots. Like when it says that a root is an induced function from R to R that maps a to O. Also I didn't understand what corollary 4.19 on the bottom of page 4.4 was saying.
2. The definition of primality or irreducibility was just like with normal integers. As was the unique factorization. The proofs were similar too. I think that this section seemed pretty straight forward and similar to what we've already done.

Sections 4.1-4.2 and App. G Due May 11

1. Long division with polynomials is very hard for me. Also the proof of the division algorithm for polynomials was a little overwhelming. Over all it seems like it will be hard keeping track of everything in these proofs because they are so big and there as so many variables.
2. I noticed that these sections were almost identical to the sections of normal integers. The division algorithm, the euclidean algorithm, definitions of division and gcd's, everything was similar to the normal steps we take with integers.

Section 3.3 Due May 9

1. Everything was familiar until the bottom of page 72 when it started talking about a homomorphism of rings. Then I was totally lost. I didn't understand what theorem 3.12 was or the proof of it was talking about. Also it seems like it will be pretty hard to find an isomorphism or homomorphism between to rings.
2. I learned most of this in my linear algebra class, but with vector spaces, not rings. As of right now I'm having a hard time seeing a real world application for this section, interesting as it is. I guess we do create bijections all the time when we pair things up and correlate things.

Section 3.1-3.2 Due May 6

1. For some reason I'm still confused about what an integral domain is, and how if it is finite, it is a field. In theorem 3.6 it says that a subset of a ring is a subring if it is closed under subtraction and multiplication, but before it was addition and multiplication. Does it have to be all three or just either subtraction or addition?
2. The properties seem pretty usual. The exponent rules are the same as for the reals, as well as squaring a+b. None of the properties of rings seem out of the ordinary. The fact that there are many units in the matrix ring M(Z) is interesting.

Section 2.3-3.1 Due May 4

1. I didn't understand what the integral domain ring was. I was also confused by the definition of a field, because I'm used to my linear algebra teacher's definition. But maybe these two aren't the same thing. Also the example on page 43 of the set T= (r,s,t,z) was so much different then the other examples, and the addition and multiplication in the tables was defined strangely.
2. It's interesting how much these rings remind me of vector spaces. The definition is almost the same if not the same, as well as the properties. Also, a subring seems just like a subspace. Are these simply two names for the same concept?

Sections 2.1-2.2 Due May 2

1. Congruence classes have always been hard for me. Especially the arithmetic. The multiplication and addition charts are easy enough, but for some reason I have a hard time using the properties of equivalence classes in proofs. It gets so hard to wrap my mind around all the a's, b's, c's and n's. It's difficult to see which properties to use to get to what we want at the end of a proof.
2.  Well this definitely relates to changing the base of a number, and also, like I said in my last post, to sending messages through computers and to binary code. It's interesting that the associative, commutative, additive, etc. that we learned in middle school apply to not just normal arithmetic, but also to modular arithmetic.