Review Questions, due Sept 30

Which topics and theorems do you think are the most important out of those we have studied?

Division Algorithm, Euclidean Algorithm, Fundamental Thrm of Arithmetic, GCD and LMC, Congruences, Fermat's Thrm, Euler phi function, solving polynomial congruences, Chinese Remainder Thrm, quadratic residue, legrendre and jacobi symbols,

What kinds of questions do you expect to see on the exam?

I expect to see some proof of some thrms, also some computation like solving a polynomial congruence. I also expect to see a problem on quadratic residues, maybe also a question using GCD's or LMC's.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Friday.

I need to work on understanding proofs involving quadratic residues better.
Show that if p is congruent to 1 (mod 3), then (−3/p) = 1.

3.1.1, due on Sept 27

1) (Difficult) I have no idea what Lemma 3.1.1 is saying and its proof is even more confusing.

2) (Reflective) I thought the most interesting about the reading was that there were so many proofs of the Thrm: There are infinitely many primes. I didn't know there were so many proofs of this thrm.

pg 221-222, due on Sept 26

1) (Difficult)I'm not sure I know what (a/p) or something of that form means.

2) (Reflective) The most interesting thing that I read was that if p is a prime then the Jacobi symbol and the Legendre symbol are identical.

2.5.2, due on Sept 20

1) (Difficult) I had the hardest time understanding Lemma 2.5.2.1.

2) (Reflective) I thought that the most interesting thing about this section is that Thrm 2.5.2.4 uses the quadratic formula. Didn't think that I would see that in this class.

2.5.2, due on

2.5.1, due on Sept 18

1) (Difficult) The most difficult part of the reading to understand was the Chinese Remainder Thrm.

2) (Reflective) The best part about this section is that it reminds me a lot of previous math classes that I have taken like Calculus or below. It reminds me of these classes because it gives a thrm or two and proves them and then uses them to find the solution to an equation.

2.4.5, due Sept 15

1) (Difficult) I had trouble follow the proof to Thrm 2.5.1.

2) (Reflective) After taking Abstract Algebra this last spring, I was pretty familiar with cyclic groups, so this section wasn't too difficult to understand. It was nice.

2.4.4, due Sept 13

1) (Difficult) I had trouble understanding Thrm 2.4.4.5 and I don't know that I fully understand Thrm 2.4.4.6. I don't understand for example how the number 7 in their example can be 7=2,4,p^k, 2p^k for p a prime. I know 7 has primitive roots, but I do not see how that statement is true about seven. I also find it would be difficult to find the primitive roots for a large number.

2) (Reflective) I have seen some of these thrms and proofs before. I am familiar with Lagrange's Thrm and Fermat's Little Thrm. I am also familiar with cyclic groups, abelian groups , and order.

4.2.3, due Sept 11

1) (Difficult) I had trouble understanding what a reduced residue system was and also had a hard time understanding Lemma 2.4.3.4 and Lemma 2.4.3.5.

2) (Reflective) I notice that there were some thrm that didn't quite make sense to me when I first read them, but through reading the proof for the thrm I was actually able to understand what the thrm was saying. For example I was able to do this with Thrm 2.4.3.1.

2.4.2, due on Sept 8

1) (Difficult) I had the most difficulty understanding the concept of a ring of integers modulo n. I think I might need an example in order for me to wrap my head around it.

2) (Reflective) The most interesting thing that I read is that in Thrm 2.4.2.2 the residue class for x was the multiplicative inverse.

2.4.1, due on Sept 6

1) (Difficult) I had trouble understanding what a residue system was. Consequently, I struggled to also understand the lemma associated with residue systems, Lemma 2.4.1.1.

2) (Reflective) Though it was hard to understand the concept of a residue system, when they talked about a residue class, I'm fairly certain that a residue class is also referred to as an equivalence class. If they are the same, then I am most familiar with the term equivalence class ([x]).

2.3, due on Sept 2

1) (Difficult) I struggled to understand Thrm 2.3.4. I have no idea what a noncommon prime is. I also didn't understand the notation.

2) (Reflective) I can connect this material to previous things I have learned. It seems I have seen a lot of these proofs before, like the proof of Lemma 2.3.1, Thrm 2.3.1, and the Fundamental Thrm of Arithmetic. I like seeing things that I have seen before because I confirms that all mathematics is connected. It is not disconnected like some people might think.