9.5 and Final, Due April 11

Difficult: It is difficult for me to understand when something is isomorphic.

Interesting: I think it is interesting that they had us prove Corollary 9.31 in the homework of 9.3 and now here it is as a corollary in 9.5.

Which topics and theorems do you think are important out of those we have studied?
Cauchy's Theorem, First Isomorphic theorem.  Pretty much the ones that were on the midterms.

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.
I definitely need a big review just on every thing that we did at the beginning of the semester with rings and fields.

What have you learned in this course? How might these things be useful to you in the future?
I have learned how important it is to know how things came about in mathematics.  This will be helpful in the future when I am teacher and have my own students and I will be able to study more the mathematics I am teaching and give them a proof of how/why that particular math concept is.

9.3 and 9.4, Due April 8

Difficult:  What is the difference between the centralizer and the center of a group?

Interesting:  The Third Sylow Theorem can be used to determine if a group is simple or has proper normal subgroups.

9.2, Due April 6

Difficult: theorem 7.9. What are the k and the t?

Interesting: The fundamental theorem of finite abelian groups.  Every finite abelian group can be written as a direct sum of prime cyclic groups.

9.1, Due April 4

Difficult:G_i is not a subgroup of the direct product G_1 x G_2 x...x G_n.

Interesting: Direct products are not commutative.

8.5, due April 1

Difficult: For theorem 8.26 why is true for all n except for 4.

Interesting: A_n is the product of 3 cycles.

Midterm 3, Due March 30

Most important: First Isomorphism Theorem, knowing how to prove something is a group and subgroup.

Questions to expect: Proving that something is a normal subgroup, finding kernels, definitions, examples.

Need work understanding: I struggle with coming up with examples for things and being able to explain why.  I also struggle with knowing where to start with proofs.

8.4, Due March 28

Difficult: Why a third isomorphism theorem? What is the point?

Interesting:  Definitions are all pretty much the same just different notations so that you can recognize that you are talking about a group.

Carl Simon: Complex Systems

Difficult: He was talking a lot about like sicknesses and stuff so it made things kind of hard to follow just because I don't know a whole lot about that stuff.  Also how can you measure/ write an equation for how contagious something is?

Interesting: Mathematics is an essential part of solving just about any problem. Everything is a system and you use calculus for just about anything things just need to be kept simple.

8.3, Due March 25

Difficult: I was noticing that |G|/|N|=x then G/H is isomorphic to Z_x.  Is this always true.

Interesting: That (Na)(Nb)=Nab

8.2, Due March 23

Diffucult: I am confused by the note where it says that a^(-1)Na=N does not mean that a^(-1)na=n for each n in N, all it means is that a^(-1)na=n_1 for some n_1 in N.

Interesting: It was interesting to see it mentioned that just because Na=aN doesn't mean that a commutes. That is something I was noticing in the homework.

8.1&8.2, Due March 21

Difficult: So every finite group is isomorphic to exactly one group?

Interesting:  It is interesting that you can take one subgroup from a group and it not be normal and then you can take another and it is normal. Also interesting that just because Na=aN doesn't make the group abelian, but it works the other way around.

8.1, Due March 18

Difficult: How you can have something modulo a subgroup.

Interesting:  The parallel of congruence with integers and rings to groups.

7.5, Due March 16

Difficult: I don't understand how this idea of cycles connect to what we have been doing.  Like why is it important?

Interesting: Every transposition is it's own inverse.

7.3, Due March 11

Difficult: There is so much notation that was used one way in earlier math classes and I feel like in this class we are using it in a completely different way.

Interesting: If something is a subset of a group then it is a subgroup if that group.

7.2, Due March 9

Difficult: So just to be clear ab doesn't mean multiplication we are just being lazy and not writing the *? So is a^2=aa, would that be multiplication or is that still what ever operation we are performing on the group? Also this is from the previous section but is order the number of element in a group or is it something else?

Interesting: It's interesting that an element can have order.

7.1, Due March 7

Difficult: In the third edition book what is the point of having a 7.1 and a 7.1 A.  They were essentially the same except 7.1 had theorems and 7.1 A went into more depth on the examples.  So some of the examples in 7.1 were hard to understand but looking at 7.1 A cleared things up.

Interesting: It's interesting that if you take the full ring of Z, R, Q, etc that it is not a group under multiplication but is under addition.  It's just interesting that when you change the operation it is no longer a group.

7.1, Due March 4

Difficult: I'm not quite understanding how example 5 is a representation of a group.  Is it just each rotation represents an element of a group? The *, can that be any operation of your choosing or is it always the composition?

Interesting: It is nice that a group is defined by one operation.

Midterm 2, Due March 2

Most important: Ideal, irreducible/reducible polynomials, showing that polynomials are fields, integral domains, or rings, First Isomorphism Theorem for rings.

Questions expected: Definitions, examples, state and prove a theorem.

Need to work on:  I am very unfamiliar with all of the theorems. There is so much to know and it's not easy stuff to just memorize.  I am not good at proving things.  I just don't know where everything comes from.

6.3, Due Feb 29

Difficult: I am not understanding example one.  What would be in P=(6)?

Interesting: That an ideal may have more than one maximal ideal.  Because when I read maximal I think maximum, and most things don't have more than one maximum.

6.2, Due Feb 26

Difficult: Saying that something is a homomophic image is still a bit confusing to me.

Interesting: It's interesting that pi is a special homomorphism the natural homomorphism, but what makes it so special.  Why can we say that is true for pi but nothing else?

6.1,6.2, Due Feb 23

Difficult: There are so many new terms and it is getting really overwhelming and I am having a difficult time seeing how they really all differ because all of the rules seem to hold for whatever we are talking about. Why do you need a special name for a+I?

Interesting:  I found it hard to find something interesting because I was so lost in the reading.

6.1, Due Feb 22

Difficult: I feel like an ideal is the same thing as saying something is commutative. What is the difference between the two?

Interesting: Proving that something is an ideal doesn't seem to difficult to understand.

5.3, Due Feb 19

Difficult: The term extension field is a little bit confusing to me. Is the extension field the roots of f(x)?

Interesting: It is interesting yet still confusing to me that a root can be [x], because to me that is a variable and I don't see how that can be put in the place of a variable to call it a root, because I feel like you are just back at the beginning.

5.2, Due Feb 17

Difficult: I'm still not quite understanding how to find the congruence classes of a polynomial.

Interesting: It's interesting that polynomials can have units and identities as well.

5.1, Due Feb 16

Difficult: I'm still a little bit confused as to the relationship between congruence classes and remainders.

Interesting: It's nice that the properties for congruence classes carries over from the integers to polynomials.

4.5 and 4.6, Due Feb 12

Difficult: When it says that A polynomial is irreducible in the complex polynomials if and only if it has degree 1, is that essentially saying that every polynomial in the complex polynomials is reducible except for the ones that are degree one?

Interesting:  I like Eisenstein's Criterion. It makes deciding whether a polynomial is reducible or not very easy. But it's annoying that it doesn't work in mod n.

4.4, Due Feb 10

Difficult: Are roots and remainders of polynomials the same thing?

Interesting: It is interesting that you can say if a polynomial is irreducible it has no roots.  I mean that makes sense, but what is interesting is that a polynomial can be reducible but still have no roots.

4.3, Due Feb 8

Difficult: Just to be clear, the associates of a polynomial are the essentially like units in the integers?

Interesting: Polynomials can be "prime" we just call them irreducible.

4.2, Due Feb 5

Difficult: I am just a little bit confused with what it says in example 2.  I agree that x^2+1/3x+2 is the gcd of that polynomial.  But what is confusing is that it says that it is "a" gcd of the polynomial.  Does that mean that there can be multiple gcds of a polynomial?

Interesting: multiples of polynomials that divide a polynomial also divide the same polynomial.

4.1, Due Feb 3

Difficult: I'm a little bit confused as to what the elements of R[x] are.  Are they the coefficients of the polynomial or are they the values of what x can be?

Interesting:  We are using all of the properties of polynomials learned in high school and then just using them to determine fields and integral domains and such.

Midterm 1 Review: Due Feb 1

1. Most important: the stuff about rings, fields, integral domains.
2. I expect to see question like what is this theorem and how do you prove it.  Proving that something is a ring or a field or a integral domain.  Giving examples of such things.
3. I need to work on knowing how to construct proofs.

10.4, Due Jan 29

Interesting: I understand how the relation ~ works, like the concept behind it and how things are equivalent.

Difficult: I do not understand how the idea of ~ relates to everything else in the section.

Due Jan 27

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

I have spent between an hour to 2 hours on homework assignments.

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

It has been really helpful to do the reading before hand that way in class I can really focus on the parts that I didn't understand in the reading and not have the lecture be the first time that I am learning it.  I do also use the internet a lot for help.

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.)

For myself I should probably go to office hours and get help from either you or the TA when I have questions.

3.3, Due Jan 25

Difficult: The relabeling part was confusing to me.  Can't you just take any two sets and relabel one to match the other and it will work.  I guess I don't understand the purpose.

Interesting: Isomorphism and Homomorphism kind of seem like proving something is one to one or onto just with different names and maybe a little bit more properties.

3.2, Due Jan 22

Difficult: Not really understanding why we can't just subtract.  I mean isn't subtraction and a negative the same idea?

Interesting: It's nice to see that the stuff we learned about units and zero divisors also applies to rings.

3.1, Due Jan 20

Difficult: The example 20 was hard for me to follow, but maybe that was just because I was unsure what the Z[sqrt 2] meants.

Interesting:  It is very nice that in order to prove something is a subring of a ring you only need to verify four of the axioms.

3.1, Due Jan 15

Difficult: I am confused because the definition of a ring is a nonempty set R equipped with two operations and in most cases that is addition and subtraction, but then there is the commutative ring that looks like it just has to satisfy multiplication.  So what is necessary for something to be a ring and can something be called a "ring" in general if it is just commutative?

Interesting: I thought it was interesting that the idea of rings can be applied to things that may not be numbers or number classes.

2.3, Due Jan 13

Difficult:  What is going to be difficult in this chapter is remembering that they aren't going to be using brackets when talking about classes.  I think it is going to cause a lot of confusion for me.

Interesting:  It's interesting to see the comparison between just regular integers and the classes.  For example when we say 2*3=0 that is not true in the integers because neither 2 or 3 is 0 but it is true in the classes because the class of 6 is 0.  So in order for the multiplication of integers to be 0 one of the integers has to be zero but for classes they can both be non zero and still equal 0.

2.2, Due Jan 11

Difficult:  It is still unclear to me how you find the classes so it is making it hard for me to see what you get when you add classes and when you multiply classes.

Interesting:  It is nice that the properties of arithmetic hold true for the classes as well.

2.1, due Jan 8

Difficult:  The whole mod n concept.  It says that a and b are congruent if their difference is 0 and equivalent if their difference is a multiple of 0.  Doesn't that make everything equivalent to each other? What is the point of the mod n.

Interesting: It's nice that the reflexive, symmetric and transitive properties hold true for congruency.

1.1-1.3, due Jan 6

What was difficult for me to grasp was the primes and unique factorization chapter.  I also am just unsure how to know when you do a proof by contradiction or need to prove it two ways.  I know that kind of goes back to Math 290 but it has been a while since I have taken that class.  Also from the reading it seems to me that 0, +1,-1 are not prime or composite. Is that correct.  That's something I have never been really sure about.
Something that I found very helpful was the way to find out if a big number is prime with out dividing it by every number between 0 and itself.  The one where n has no prime divisors less than or equal to the square root of n.  It could have come in handy in earlier math classes.

Introduction, Due on Jan 6th

I am in my fourth year of college and I am studying Mathematics Education.  I have taken Math 290, Math 313, Math 314, and Math 334.  I am taking this class because it is a requirement for my major.  The math teacher that was really great was one of my high school math teachers.  His name is Mr. Barkdoll and I had him for two years.  He really cared about my success in school and I was able to go to him for help in deciding which classes to take and which ones would benefit me at the time.  He believed in me.  He also took the time to get to know his students and he made mathematics fun with different games that continued throughout the year.  One of my favorite professors at BYU was Professor Villamizar.  He really cared about how we did in class and would come and talk to us and tell us what we could do to improve.  He made himself available and he new me by name and didn't forget me after the semester was over. A little bit about me, I am from Arizona, I am the oldest of 5 girls, I love singing, and being out doors hiking and camping. I am available for the hours provided.