Due January 28

I think the most important things we've learned so far are the theorems involving modular arithmetic, especially those concerning solving modular algebra equations, and the concepts of isomorphism and homomorphism along with the axioms of showing whether two rings are isomorphic or homomorphic.

I imagine that the test will mostly include equations to solve in modular arithmetic and potential rings to test and figure out if they have identity or are commutative or integral domains or fields or isomorphic to another ring.

3.3 Due January 23

The hardest part of the reading for me was visualing some of the examples. For example, thinking of homomorphic rings that aren't isomorphic was a little difficult at first until I realized that the image of the function is a subring of the codomain and that the function is either injective or surjective. Also, I still can't think of any properties that are not preserved by isomorphism.

I think that the concept of isomorphism will make working with rings easier, especially rings constructed from finite sets. Because if the set is finite, there must also be a finite number of distinct, non-isomorphic rings from sets of the same order.

3.2 Due January 21.

So the hardest part for me in this reading was working out the proof that while every field is an integral domain, only finite integral domains are necessarily fields. But I do find it interesting that in order to determine whether a ring, finite or infinite, is both an integral domain and a field, all one has to do is show that it is a field.

As far as reflection on the reading goes, I just think it's really cool that subtraction and exponential operations work on rings. Also, in thinking about division on rings, I think that division would be defined on a ring with multiplicative inverses.

3.1 Due January 16

Oops, I forgot to blog for last time. Oh, well. Now, this last reading was great. The most difficult thing for me was probably just checking all the examples to make sure they fulfilled all the axioms. That took a while.

What I found interesting is that the set of all 2 x 2 matrices of the form listed in one of the examples forms a field while the set of 2 x 2 matrices. This makes me wonder what other forms of 2 x 2 matrices farm fields and also how one would form a field with 4 x 4 and 3 x 3 matrices using the standard definitions of matrix addition and multiplication.

2.2 Due January 12

Well, this reading was slightly more difficult on a conceptual level than the previous readings. It was a little difficult to wrap my mind around the idea of adding and multiplying whole congruence classes instead of integers.

Over the Christmas break and in preparation for this class, I started reading up on group theory. One thing that I noticed in reading section 2.2 is that addition and multiplication of congruence classes can be used to define a group on Z_n.

2.1 Due January 9

Once again, the reading was strictly review of material from Math 190 and I've done a lot with modular arithmetic outside of 190 anyway, so I feel very comfortable with concepts in today's reading. I didn't feel challenged at all, but I did come up with a question. The book showed that that addition and multiplication still work as normal in modular arithmetic, but I wonder if the same goes for exponential manipulation. I plan on exploring this question.

I was impressed on how the use of modular arithmetic could make some of the exercises of the last section much easier. For example, one of the exercises in 1.2 was to prove that a positive integer is divisible by 3 if and only if its digits add up to a multiple of three. This proof would have been much easier with modular arithmetic because it's clear that powers of 10 are congruent to 1 mod 3 and thus it's easy to show that the sum of the digits of any number is congruent to the number itself mod 3.

1.1-1.3, due on January 7

To be honest, I wasn't really challenged by the reading assignment. I have just recently completed Math 190 so all of the reading was review. However, it did take me a little time to completely understand the proof of the division algorithm.

I love the idea of unique prime factorization. I'm really impressed on how it has increased my understanding of algebra concepts, especially those concerning divisibility. And understanding it really helped me on the exercises of the earlier sections, interestingly enough.

Introduction, due on January 7

What is your year in school and major? I'm a sophomore and a mathematics major.
Which post-calculus math courses have you taken? Math 190, Math 214, Math 334 and Math343.
Why are you taking this class? (Be specific.) For my major, but more importantly because it sounds really cool.
Tell me about the math professor or teacher you have had who was the most and/or least effective. What did s/he do that worked so well/poorly? I think Dr. Doud here at BYU has been the most effective teacher that I've had. He did everything in a very orderly manner and always stayed true to his word. Also, he presented things very logically and clearly.
Write something interesting or unique about yourself. Well, I've worked three summers as a river guide on the slmon river in Idaho. Also, I served a mission in Albania and now teach Albanian at the MTC.
If you are unable to come to my scheduled office hours, what times would work for you? Any time on Tuesdays and Thursdays.