Due on September 30

I think the most important topics are things like quadratic reciprocity and knowing how to "unpack" definitions.Also, theorems that tell you quicker ways of doing things or thinking about things. I'm hoping that the questions on the exam will be similar to 371 tests where we'll have computations, some definition type questions, 1 or no proofs. I need to work on my understanding of the Chinese Remainder Theorem, I'm still unsure of exactly how and when to use it. I would like to see more non-trivial examples of this in class.

3.1.1, due September 28

In order to prove that there are infinitely many primes, I most like the first proof, which uses p1p2...pn +1. I like it because it is concise, proves the point, and makes sense for my way of thinking. The next proof is fine, but then I don't really understand the proofs that use factorials. I don't really like have so many different proofs of the same theorem because I'm the kind of person who likes to pick what works for me and then "not worry" about the rest. I feel like if I take that approach with this section, then I'm not learning what was intended.

Jacobi Symbols, due September 26

I think this will be a great section to talk about in class because the concepts weren't overly confusing, but the lecture will help me better understand the connections. Also, from now on, when we write (a/n) (and the context makes sense) is that just a symbol for (a/p1)^e1...(a/pk)^ek? The only thing I really see the Jacobi symbol useful for right now is as the shorthand way of writing the above, or for use in definition 5.3.1.4. Is it helpful in other ways? I kind of like the whole pseudoprimes just because I feel like the name actually kind of hints at what they are.

2.6, due on September 23

First, it is important to note that I will not be at your office hours tomorrow! Not because I can't make it or because I don't want to come, but because I have already finished the homework assignment and I understand all of them! One thing I need to do is get a little clearer on my definitions, especially of quadratic residue, but also some older definitions that are make it hard to understand new things. I think lemma 2.6.1 is cool because it tells us how to quickly find or eliminate quadratic residues which, i think, are basically solutions. (maybe i'm not understanding) But if this does what I think, then it's a pretty sweet lemma. I don't really like using the "Legendre symbol" to mean quadratic reciprocity law because then it just feels like an extra two terms that I need to memorize.

2.5.2, due on September 21

I thought I understood the Chinese remainder theorem, but when they used it to find the solution to example 2.5.2.1, I wasn't sure where they got the numbers that they were multiplying together. The theorem that says if f(x) has degree higher than p, there exists a polynomial of degree less than p where the solutions to the two functions are the same is a little bit hard for me to believe. I just can't think of why this would have to be true, but if i become convinced that it is, then it's a pretty cool concept.

2.5.1, due on September 19

As soon as we start working with polynomials modulo, it becomes a bit more challenging for me. I don't feel like this section will be too difficult but I am little weary of getting into the higher degree polynomials. I like this section because things still work mostly in the ways that we've already learned. It does seem like there are more "rules" to memorize, but hopefully that is because I haven't really learned it yet and once I do, the rules will make enough sense that they won't feel memorized.

2.4.5, due on September 16

It's nice that we just talked about cyclic groups and that they were near enough to the end of abstract algebra that I remember them better than a lot of things. It's cool how we can use algebra to prove number theory as well as using number theory to prove algebra. The subjects are very interconnected if we choose to look at these relationships. I find this helpful since I did understand most of the things from abstract algebra, so when I can rely upon that knowledge, it makes it easier to learn the new things. The hardest things from this section is that I know I understood these theorems and could remember all of the "rules" at one point, and so it's frustrating when I can't remember them in as detailed of a way as I would like to.

2.4.4, due September 14

I don't think I am understanding Euler's theorem and primitive roots. I think it's because I was just beginning to understand the phi function and how to use and and I haven't quite finished wrapping my mind around it in order to utilize it in this new context. I do like the method for finding primitive roots (even if i don't completely get them) because it is concrete and I like having one solid way of doing things every now and then.

2.4.3, due on September 12

The Euler Phi function idea was a bit confusing to me, but I think that is mainly because I've only read it and this book isn't the best at explaining things. So I expect I'll understand a lot better once I've heard it explained as well. I think that will make it a lot more concrete. I still loving getting to relearn things we did in 371 because it makes them so much more solid in my mind to see them looking from the perspective of already having a fairly good grasp on them.

2.4.2, due on September 9

I feel like there was a simpler way of finding multiplicative inverses in Z modulo n, but I can't remember what it was. The euclidean algorithm and writing 1 as a linear combination of a and n makes sense, I just thought I remembered a shorter way of finding these inverses. I do feel like after reading the proof again, I better understand the process of finding multiplicative inverses and why it works. With Wilson's Theorem, I understood about the first half and then it seemed like they dropped one proof and started another so I got confused. I also didn't understand why Wilson's theorem is useful.

2.4-2.4.1, due on September 7

I'm still having a difficult time with the different language that this book uses compared with the one that we used for Abstract Algebra. I also feel like this book is less particular about things which has, so far, only confused me. Aside from that, it is interesting to look at congruence classes again and hopefully I'll quickly remember what I learned so that I can enjoy it again. I'm not sure if I quite understand what they mean by residue classes, sets of residue classes, complete residue systems...is this just different vocabulary for something we've already learned?

2.3, due on September 2

One thing I am a little confused by is the way that this book proves that any integer > 1 can be expressed as a product of primes. I seem to recall the other book using a different proof which I thought made more sense. Perhaps that is because I don't typically use induction proofs, I think I should hurry and get over that. Another thing that confused me is when they did the prime decomposition and they have 67^0 and also how they used the prime factorization in order to find the GCD and LCM. I didn't know that if an integer is not a perfect power then its root is irrational, so that's kind of neat.