Wednesday October 2 Section 7.2.

1. dont understand the part in the proof of theorem 7.5 how by theorem 7.4 would be a power of 2.

2. seems like a very useful way of determining the impossibility of proofs. any shape that requires numbers other than square root of 2 cant be draw with the ruler and compass construction. 

Monday September 30 Section 7.1

1. is the p1 and p2 in figure 7.2 suppose to be the center of the circle?
the Ki notation in this chapter is really hard to understand.

2. its really interesting how the construction of graph can be linked to algebra

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?

just over an hour on average. Lecture helps the most in doing the assignments, but the reading beforehand do help me make sense of the lecture. 

• What has contributed most to your learning in this class thus far?

lecture and homework. office hour is helpful too. If possible, I would like to have extra office hours during this next week so that I can get extra help while I prepare for the midterm

• What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)

I would appreciate more office hours. Also, having a review sheet for the midterm and some past midterm sample to practice on would be really helpful.

Careers in Math 26 Sep Thur

the Career in Math talk didn't really talk about any difficult problems, so I would just write down a few things I found interesting about it.


The talk is about Raytheon, a company that develops innovative technology for National defense.

the speaker emphasize that the problems that they are solving is often very difficult and involve high level math skills. Going to graduate school is encouraged since that gives you much more skills in those problem solving.

they just developed a technology for crowd control. It sends out microwave of certain frequency that will not go pass our skin so wound't cause any damage to body, but makes people under the wave feels uncomfortable, thus able to evacuate a crowd.

Friday September 27 Section 6.2.

1. this chapter's idea is rather easy to understand. the proof of short tower law is a bit hard to follow with all those subscript, but its easy to see how the short tower law works in real example.

2. interesting to see that algebraic extensions need not be finite while every finite extension is algebraic.

Wednesday September 25 Sections 5.4 and 6.1.

1. How is the Lemma 5.14 a restatement of Lemma 5.9? I don't see how the basis in Lemma 5.14 relates to Lemma 5.9

2. interesting to see that the complex number are two-dimensional over the real numbers.
It is obvious to see that the degree of real numbers over the rationals is infinite; what about the degree of the real numbers over the irrationals? would it be one?

Monday September 23 Sections 5.2-5.3.

1. in theorem 5.7, we require the alpha to be algebraic over K just to ensure there exist a polynomial giving a root of alpha? or for some other purpose?

2. modulo rules applying to polynomials looks very useful, since we have developed so many theorems on modulo arithmetic already

Friday September 20 Sections 4.2-4.3 and 5.1.

1. the notation in Definition 4.12 is hard to understand.

2. So Example 4.11 has demonstrated that i + sqrt(5) is enough to generate Q(i,-i,sqrt(5), -sqrt(5)). but with other situations, like if I have a (i)^(1/3), how do I tell if something is enough to generate all the needed elements?

the extension example shows that pi has a different type of irrationality than just square root of like 2 or 3?

Wednesday September 18 Section 4.1.

1. why is a monomorphism being an one to one mapping would maps from a small field to a large field? (K to L) (p.50)

2. so far it seems like in finding Q(X), we only need to try out the possibilities in multiplying some elements to X to get all the possible forms that it generate. Are there examples where we need to look at the inverse also to include all the possible forms the X could generate?

Sections 3.5-3.6 (due Mon Sep 16)

1. the example in 3.23 is a bit confusing. I understand why t^4+2 is irreducible over Z5; but when they go to Z3 suddenly they found a factorization of the polynomial and concluded it reducible. So how do they come up with the factorization in Z3?

2. How is finding the multiplicity of the zeros useful?
in p.45 it points to figure 1.1 when explaining the multiplicity, but I can't find the right figure 1.1

3.2-3.4, Due Sep 13

1. Example 3.13 about proving irreducibility with Theorem 3.12 is a bit hard to understand.

2. The eisenstein's criterion is an interesting way to show an polynomial irreducible.

Sections 2.3 and 3.1 For Sep 11

1. why is the factorization unique in the Division Algorithm proposition 3.1?
its easy to see in real numbers, but how do we know if we cannot expressed the polynomial in another form with complex number?

2. Its interesting to see the extension of prime in integers to polynomials in theorem 3.9                           would the same method be used in finding how to express the hcf into f and g?

1.0-1.4, due on September 6

1. I am still not very familiar and comfortable with using omega (the primitive cube root as defined in the book) to express the multiplicative inverse (like in homework 1.5, I still couldn't think of ways to express 1/(p+qa+ra^2) in the x+ya+za^2 form)

The Cardano's Formula part is a bit difficult to follow.

2. A side question that comes to me while reading page 5: do theoretical Mathematicians (as opposed to Applied Mathematicians that study Math problems with rather immediate practical usage) makes significantly less than Applied Mathematicians?

Also, the existence of Complex number has made much more sense to me when the book explain it from the prospective of the need of solving more and more difficult polynomial equations, how we expanded from N to Z to Q, R and eventually C so that we can expressed the roots of the polynomials. I am interested to know if there's a set bigger than C that is required to solve some equations.

Introduction, due September 6

1. I am a Junior Math major.

2. I took Math 314, 334, 341, 371

3. I am quite interested in number theory, in fact I am hoping to join a number theory research group soon. 

4. Dr. Lawlor is the best. He thinks about how to present the lesson so that we can easily remember the important theorems; He has lots of office hours so I never felt hindered to go asking him questions and most of the things that I am confused during class were answered during that time. Most of all, I see a real sincerity from him in teaching and helping students.

5. I am from Hong Kong, served mission in England. I play Chinese Chess and Go a lot. 

6. The current office is not a problem for me but I would appreciate if there can be office hours on Tuesday and Thursday too (like Tuesday and Thursday around 2 or 3pm)