LAST EVER BLOG POST

Which topics and theorems do you think are important out of those we have studied?
Probably none of them.  I don't care anymore
What do you need to work on understanding better before the exam?
everything from chapter 8...
Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

What have you learned in this course? How might these things be useful to you in the future?
Groups, rings...proof writing, analytical thinking.  problem solving.  reading a textbook cover to cover and working examples.

I value these skills, but have no current application for groups/rings right now.  I will be teaching seminary...so I'll put abstract algebra in the back of my mind until...

8.5 Due less than a week away

Difficult)))

not really anything in this reading was hard.

It was also not very enjoyable...kinda a no-brainer...

structure of finite groups...feels right.  Good stuff.  Probably useful somewhere, but right now I'm just done with the semester, and I don't really care about this topic anymore.  I need some application.

Reflective)))
Seems like a pretty good summation of the semester...bring on the final.

8.3 - 8.4 Due April 7

Difficult)))
Sylow subgroups, finite nonabelian groups...1st-3rd sylow theorems...

not really difficult.  read...good enough.  Bored out of my mind with this class...

Reflective)))
My 9 year wedding anniversary is this thursday...I'm not going to do this blog.

8.2 Due April 5

Difficult)))

Reflective)))

8.1 Due April 3

Difficult)))
caring about a blog that offers marginal points and nothing substantial to my life or my happiness...
especially over conference weekend...when the teacher said that there is no homework for his class this weekend...oops...forgot about the blog...



Reflective)))
I used direct products of groups in Theoretical Analysis for Math 344 fall semester of ACME.  They didn't do near as well a job in their book or describing the notation...

7.10 Due March 31

Difficult)))
Ah no big deal, just one thing really...theorem 7.52


Reflective)))
I feel so much better after studying for an exam.  Regardless of how I do on the test, I really feel like I understand the big picture and see clearly how all the little things fit into place much better from the content when I condense the entire section into one page of notes representing all the theorems, lemma, corollarys, and definitions...along with examples and practice problems.

Exam Review Due March 29

Which topics and theorems do you think are the most important out of those we have studied?
None.  Just figure out the definitions and true-false statements, and examples...that is the only place I lose points, not on understanding or applying relevant information...

What kinds of questions do you expect to see on the exam?
Provide an example of a group...

What do you need to work on understanding better before the exam?
Examples of all the different possible kinds of groups that we could be asked to show examples of on the test...

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 Wednesday.
Examples.
Abelian
Non Abelian
Quotient Group
Non trivial subgroup
Group isomorphism for Z4 and Z2 x Z2
Kernal

Non-abelian groups: Sn, An, Dn, matrix groups.
Abelian groups: Z, Zn, Un.
An element of finite order contained in a group of infinite order.
Cyclic groups of all orders—both infinite and finite.
Groups which are not cyclic, including a (sub)group generated by two elements which is not cyclic.
A group with a non-trivial center.
A subgroup of an infinite group that has finite index

7.8 Due March 27

Difficult)))
Probably the Second Isomorphism of groups...
The third one uses the first, and the first isn't really in the book...but only kind of because I guess we did it with rings and are going to reuse it...

But seriously, the idea of the kernal and quotient groups looks challenging.  I'm really sad the book makes me write this section...and that the author didn't just copy and paste when he/they wrote it

Reflective)))
Take your pick from chapter 6...and change some variables to put it in terms for groups...too bad the book has as much information as this blog...

skipping proofs, and claiming they were done in the last chapter...

Frieze Group Theory

Complex systems: approaches to population by Studies Carl P. Simon
Founded center for the study of complex systems in 1999, and was CEO for 10 years

systems thinking:  composed of interdependent and interacting components.  Ignoring them leads to undesired consequences.
An example of DDT and natural systems.  Not worrying about the ramification of the system.

Health Systems: mistake of using the strongest medications.  Drug-resistant bacteria and viruses are an example of health systems.  IE an arms race with bacteria, more people died from bacteria than from HIV.  Careless use of medication.  Also anti-bacterial soaps: promulgating helpful and resistant bacteria.

Man-made demographic solutions:  widen the road when too much traffic...urban decay.

HOW TO APPROACH A SYSTEM:
Identify variables, draw diagrams, build a model

MODELS:  dynamics, the framework for data collection, estimate, parameters,experiments, comparison, assumptions, forecasts.

K.I.S.S. Principle  Keep it simple stupid.


simple systems:  
put down components and check the spread.  Check connections
Disease Spread:
build equations and variables



complex sustems:
Useful for when there is a rugged landscape without a clear path
Adding complexity line upon line to add realism and fact.
how to build feedback and learning into a model:
build a rule:
each rule has its specificity=strength
increase strength as the rule works
tax the rules to see if some are less useful
Realism:
mutation and crossover of rules allow to adjust and check for new scenarios.

7.7 Due March 24

Difficult)))

Quotient Group.  Fun new definition that I will need to play around with to really get a good grasp on the context.

I did not get the operation in example at the top of page 218 the first time.
I had to try the operation to verify that (Mr1)(Mh)=Md.  I thought that the coset elements are multiplied, but no it's the operator on the coset that forms the product.

Example  on top of page 219 I follow the example, but at the end of the example there is a brief explanation on the fact that Z not being an ideal of Q makes Q/Z not a quotient group...???
WHAT????  ^
Oh...my bad...YES a quoteient group...but not a Quotient ring.  DUH..

Reflective)))
Na=Nc and Nb=Nd => that Nab=Ncd
also theorem 7.36 is a good reminder of rings and ideals and their properties.

7.6 Due March 22

Difficult)))
when a chapter is split up...and I'm supposed to do homework without having completed a reading or a chapter...that's really difficult...I gave up and read this for the last homework...sure made everything much easier to prove...

why doesn't aN=Na => that an=na....what's the difference...?
I see...its the modulus and that there is some t in N such that na=at and at is in aN...and vice versa to obtain the right coset from the Left coset

Reflective)))
The rest of the reading for 7.6 looks a lot like the topics taught Monday in class except for this new theorem...which makes perfect sense...and sure was pretty useful in proving the problems on the last homework set...too abd it wasn't part of monday's reading...

7.5 & 7.6 Due March 20

Difficult)))

Everything theorem 7.28-7.30
I am so lost...is this due to the concept of cyclical order less than order of group??

I'm praying desparately for the explanation of those proofs monday...

Reflective)))
Left coset...kind of like right coset...but left...all the same definitions.




If you think this blog isn't long enough, then go back and read the last one again...but switch definitions from right to left...

7.5 Due March 17

Difficult)))
Why is there only a right coset of groups?  not a left? I read the note about how rings being abelian give us a + i = i + a and so a + I for rings, but I don't understand the correlation between that and why groups have Ka or K + a...and not a + K...
Oaph!  I get it...transpositions, and compositions... they are set up as the (ab)(cd)(ef)... order matters for groups!

|G| = |K| [G:K]  means that the order of K divides the order of G...  I had to look at that one twice to get that from the theorem...I feel so dumb...

Reflective)))
7.22...7.23....7.24 what great comfort this sentence gives...no need to blog...just sit back and reflect on those sweet theorems regurgitated...

7.9 Due March 16

Difficult)))
cruel and annoying notation to memorize.  I struggled so hard to see how (14)(13)(12) = (1234)  until I wrote it out in the original notation of permutations and did the composition...but that is hard to make the transition.  would have been easier to start with this new notation...

I was really confused if the book meant to say transposition on page 233 instead of transformation...in the paragraph under the bolded heading "The Alternating Groups"...they have not said anything about what a transformation is...??

Transformation is not in the book defined...I'm pretty sure they used the wrong word...annoying to read a defective text...

Reflective)))
the factorization of permutations is either even or odd...just like integers are even or odd.  Number theory permeates all of math!!

7.4 Due March 13

Difficult)))

Love the properties and tools of isomorphism, but the automorphism is still unknown as to its usefullness or value...also lost as to why the inner automorphism is valuable...

maybe since it is only in a small example, it isn't really important or useful...so I should just let it go...?????

I also struggled ot understand the corollary for 7.21...this section will require a lot more reading before class, and while trying to do the homework.

Reflective)))
Proofs of isomorphism and homomorphism are fun.  give me those any day!

a lot of these theorems are basically a rewording of the ones for rings...

7.3 Due March 10

Difficult)))
I struggled to understand the relevance of the Center of a Group, until I realized that it is used to form a subgroup...wasted time...that is always difficult.

...like wasting time typing up a blog and trying to find a justification of content that is deserving of points...like that is all school seems to be is a justification for points...

I hate grades...so finite and useless...also a difficult thing for me to put up with...



Reflective)))
ring, subring.  Field, subfield.  Group, Subgroup.  Larger set having properties, smaller group inherits similar properties.
Cyclic and generators are definitions that I have seen used, and had examples of before in other clases, but never understood the full application of the definition until this class...in the future, when I understand it more than after just the first reading.

7.2 Due March 8

Difficult)))

I have NO IDEA how the example of order of an element of S3 works?  what is the operation of raising S3 to a power?????????????
HA!
It's a composition of permutations.  3->2,2->1 and so 1->1 ect...so the exponent on a permutation is the definition of that permutation composed of itself that number of times.

Let G be an abelian group in which every element has finite order.  If c in G is an element of largest order in G (|a| <= |c| for all a in G) then the order of every element of G dividec |c|....
But what does that even mean???

Reflective)))
I remember being in Trig and being told that i^4 = 1...and that it is a cyclical group...and having no idea what that meant...Coooooooooooll.

7.1 (part 2) Due March 6

Difficult)))
Abelian definition (I used that complete definition while studying for the last midterm and defining the properties of a ring.)
The hardest part of the group theory reading from this section is probably just remembering that the multiplication sign is + for Z,Zn,Q,R,C...and not multiplication because a ring under usual multiplication is not a group unless the ring is the zero-ring.


Reflective)))
I remember When I took complex analysis and we did composition of functions and linear mappings, it seems to be a lot like the bijective composition of functions that we also used in algorithm design when the class covered fast-fourier transforms of compositions of functions.

7.1 due March 4

Difficult))) (balancing time with a CS project that has taken too long and a homework load for every other class.)
Group theory looks like fun!  I don't think that anything looks hard, but when I was in the ACME version of 342 I found that  the second law of isomorphisms and the application on transformations looks a lot like the structure of groups.

I'm going to have to re-read this section to understand it better.


Reflective)))
The exam was a lot easier this time...now that I have read through the examples and had some in mind...
while studying for the exam, I read about albian groups in the properties of ring addition!!!
Happy that's over with.  Tests are fun when they aren't Computer Science exams.  Who codes without google and references...NOT ME...sorry, but blogs are for thoughts, and so I am reflectng on the semester, and what I have really learned is that I am not planning on coding for my day job...unless it is in python or R.

Exam Prep Due February 2

Which topics and theorems do you think are the most important out of those we have studied?
Factor, isomorphism, reducibility, remainder, relatively prime, prime (polynomials/ideals), congruence classes, cosets, polynomial functions, quotient rings, Kernal, Maximals...

What kinds of questions do you expect to see on the exam?
--Give me examples of X where X was discussed either in class or on a homework.
--what is the definition of Y where Y is contained in the material covered after the last exam
--Prove Z, where Z was in the reading/class lecture/homework
--some random new concept teacher finds interesting and teaches on the review day...

What do you need to work on understanding better before the exam?
--All of the above.
--Congruence class arithmetic.
--A real understanding of X as either a variable in the rules of functions, an element of a set, a number.

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 Wednesday.
I didn't follow the why of the proof of theorem 5.11 (second edition) for (alpha)=[x]  and the definition of congruence -class arithmetic.

6.3 Due February 29

Difficult)))   (aside from doing homework on the weekend with 4 kids...when I already have a career job and have no more desire to do school...no offense to rings theory)

<<COPY AND PASTE BLOG FROM 1.3>>  then add to it some random blah about maximals as ideals.

R/M is a field iff M is a maximal ideal of a commutative ring with identity.  I enjoyed reading through the proof to understand how it worked.

Reflective)))
The most difficult concept in this reading section is the understanding of the definitions of the divisor, and the application points of prime numbers.  I had to do a few simple example problems to convince myself of the fact of the Fundamental theorem of Arithmetic to verify that the statement was true.  It helped to notice that the definition of p being prime is true if and only if -p is also prime!

I really enjoy the Euclidian algorithm, and using it to find (a,b) or the greatest common divisor.  It is also very useful in knowing and applying understanding for prime numbers.  I'm excited to use these concept to describe modular arithmetic and building on these principles.

6.2 Due February 26

Difficult)))
I have to read theorem 6.10 and the fact that every kernal is an ideal and that conversely every ideal is the kernal of a homomorphism a number of times before I truly understood that idea...that was mind blowing.

Understanding the kernal and its relation to an ideal is deffinitely difficult and a powerful concept.


Reflective)))
Homomorphism of rings.  Reading through the old chapters and refreshing on the idea of a homomorphism was good study for the test.

f(a+b) = f(a) + f(b)  and all that jazz...

time...don't have it right now with a whole bunch of projects/midterms this week...planning ahead all semester, and today still too busy.

6.1 & 6.2 Due February 24

Difficult)))  the idea of a coset was a surprise to see with equivalence classes and congruence classes.  I remembered it from another math class kindof, so I looked up additional readings to help me understand it better.

That means that R is partitioned into equivalence class under this "congruent to modulo I" relation. The equivalence classes are called "cosets" ... and "the coset of x" just means "the equivalence class of x under the equivalence relation "congruent modulo I"".
Which I got from :   http://math.stackexchange.com/questions/23655/what-is-a-quotient-ring-and-cosets



Reflective)))
It's very easy to be reflective with this.  I like that the ideal has cosets and that the cosets of a ring are either disjoint or identical.  It would be much more helpful if the book actually rewrote the proof and allowed me to compare the differences.  It takes a lot more time to read through the old proof and rewrite it to fit the new definitions than it would to just read it...if they had rewritten the proof.

Number theory on cosets and the "circle plus" operator is a handy tool to have under my belt in taking this class...if only it was introduced as clearly then as it was here...should have taken this class first...

6.1 Due February 22

Difficult)))
The absorption property of an ideal seems very useful.  I like the fact that congruence and theorem 6.1 build a useful application for the use of an ideal and it will be interesting to connect the relavency in rings as I gain more understanding about the ideal.  Right now I barely understand the idea of an ideal, but with the homework and a lot more examples, I should be able ot understand it much better.

Reflective)))
It looks like we have seen many ideals in the book so far...like identity in any ring is an ideal, but the concept of a finitely generated ideal looks like a property of polynomial rings.
But I am stuck on any other relationship or value that the ideal gives us...

5.3 Due February 19

Difficult))) Every finite integral domain is a field.  This concept never did sink in previously, but as I read the details of theorem 5.10 it did.  Also that the number of elements of fields {Zp[x]/(p(x))} is p-raised to the degree of p(x).

F is a subfield of F[x]/p(x).  F[x]/p(x) also called an extension field of F and if p(x) is irreducible in F[x] then p(x) has a root in F[x]/p(x).

Eventhough p(x) is irreducible in F[x] it may be reducible in F[x]/p(x).  In fact

From there, it all kind of fell apart in a heap of confusion.  How does multiplication work, and addition, and how does C relate to R[x]/x^2 +1?

I have lots of questions and I need to see lots more examples before I find consolation in this section

Reflective)))
This section shows us how complex number systems were established and became common use.
It helps us solidify when F[x]/p(x)   is a field (therefore an integral domain).
We learn a vocabulary word of the extension field.
And there is a relation defined for a root of an irreducible p(x) in a field extension.

I am looking for teacher to provide further examples of application for the theorems in this section.

This concept was harder fro me to graps than any other.

5.2 Due February 17

Difficult)))

It would be very challenging to fill out the tables.  When we did this for section 2 it was just exhaustive...but this would actually be really challenging to think through and do the division algorithm on each element for the multiplication tables.

Reflective)))
Um...the whole section is reflective of section 2.2...does that count?
Theorem 5.7 is a very cool introduction to the isomorphism that is created with the set of the modular classes that exist for the set of all congruence classes modulo p(x) is denoted.

I'm excited to go get to work on the homework problems and to do some practice problems on the sets.

5.1 Due February 16

DIFFICULT)))
corollary 5.5 was horribly confusing to me!!!!!!!
I understand that the properties of Zn are analogous to F[x] and was able to see and follow the analogue clear enough, until this corollary.  I fail to follow the proof and am struggling to see how the set S is able to form a unique form of a polynomial while at the same time having only n distinct equivalence classes.  for instance, the example that there are infinitely many distinct congruence classes of R[x]/(x^2 + 1)  but in Zn we have n distinct congruence classes...?????
ok...nevermind.  I read through more examples, and now understand why R[x]/f(x) has infinitely many equivalence classes.


Reflective)))
This reading was slow...I'm so excited to be graduating soon, and for summer internships and freedom from tests and homework!!!  I'm struggling to enjoy reading about equivalence classes and polynomial fields and then trying to create a reflexive comment on the reading...it all made sense, and I'm going to read it again the day it will be presented, and I'll reflect then on what is important to remember for a test...so next time I'll probably write something useful in this section.

4.5 4.6 Due February 12

Difficult)))
That is insane that the rational root test and then the eisenstein criterion are true!  the proof was very hard to follow, and I think I spent about 30 minutes working through each proof to show myself why it actually worked...but it is amazingly useful!  Such detailed specifics are powerful tools for showing irreducible polynomials in Q[x].
Ahhh!!!  conjugates of complex numbers...I had to run through a crash review on how operations work over complex numbers.


Reflective)))
These would have been loads of fun to use if I had known them in differential equations!!!
The proof of 4.22 uses many theorems we have in our toolbox.  the definition of roots, factors, the rational root test, and the factorization of a reducible polynomial!!  fun stuff!

It is amazing to me that by simply picking a p such that Zp[x] doesn't divide An then having the polynomial be irreducible in mod p also guarantees the original polynomial to be irreducible in Q[x]  but I need to remember that reducible in mod p doesn't imply that it is reducible in Q[x].

BUT C[x] for any nonconstant polynomial is algebraically closed or is guaranteed to have a root and is irreducible if and only if the polynomial is degree 1.

AND FINALLY GET TO THE TOOLS I HAVE BEEN USING IN CALCULUS AND DIFF.EQU.!!!

4.4 Due February 10

Difficult)))
I do not understand the switching back and forth between an indeterminate and a variable of a function...
I do not understnd the significance or value in corollary 4.19 and why its useful that a function that maps on infinite fields is the same iff it is induced by the same polynomial??  what would make this so useful?
Seems almost trivial to try to apply the number of roots to a possibly infinite degree polynomial...??


Reflective)))
Polynomial functions are what we operate in for calculus and differential equations!  I feel so at home here working on fields of polynomials and polynomial functions defined for every polynomial in the field.  The usefullness of roots and factors, and irreducible polynomials not having roots is almost like a little safety zone for me to retreat to for a few days after struggling with the idea of X being indeterminate...but to let it be a variable that describes a rule of the function...I really feel like the stantard needs to be changed and the distinction made MORE CLEAR on when X is a variable and when it is a determinate.

4.3 Due February 8

Difficult)))
This section really helped me to clarify the understanding of what a unit is, and how to use the definition along with zero divisors to build up a concept of prime polynomials, or irreducible nonconstant polynomials.



Reflective)))
What is an example of an associate that doesn't commute.
Rather:
given "a" in a ring R where R is a commutative ring with identity where a=bu for some unit u, and some element b in R, then what is an example where b = a(u^(-1)) does not hold?????

4.2 Due February 5

Difficult))
Theorem 4.5 seems like a long and excruciating process to solve by hand the solution fo finding the form of the GCD such that GCD = f(x)u(x) + g(x)v(x)...

I'm also strugging to visualize what 1F is in terms of a polynomial.  It looks like it's just a way of describing the coefficient as unitary, but I thought it was that the whole polynomial has coefficients of 1, but that doesn't make any sence now that I think about it.

Reflective))
I love that all of the properties of rings and fields commute to rings of polynomials and fields of polynomials including the GCD, the euclidean alrogithm, division algorithm and divisors.

This is amazing, and so much fun to work with polynomials.  I always feared doing long division before because I never really understood the process or background and context, but this class is really helping me to fill in the gaps in my understanding and experience of math!!!

4.1 Due February 3

Difficult)
How much of Appendix G are you planning on us reading and understanding??  the reading only mentioned 4.1 but the book uses a heavy understanding gained from Appendix G...??

Polynomials are awesome!  I wish that when someone tried to teach me polynomial division 5 years ago, that they had used this book.  It makes so much more sense explained in detail!
The ring P of polynomials with coefficients in a subring of P, R will be denoted as R[x]


Interesting or reflective)
Synthetic Long Division!!!  WAHOO!!!
The polynomial concept of x reminds me a lot of linear independence!
If R is a sing and S is a subring of R then IS R ISOMORPHIC TO S???????????
This question comes out of the first theorem of 4.1 where we are shown that R is a subring of P where P contains a copy of R isomorphic to R plus some element x not in R.

?
?
???
??????
?????????????
?????????????????????????????
??????????????????????????????????????????????????????????????????????????????????????

study guide /due February 1

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

proof of a ring, and integral domain, and field (differences and similarities

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

prove that something is a subring, prove that something is a field, prove that something is an integral domain, show something is an isomorphism and show something is a homomorphism

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.

show the differences and similarities between a field and a field of quotients.

9.4 Due January 29

Difficult)
Lets go with all of it.  Notation, reasoning and logic of a field of quotients...deffinitely struggled with that idea.  Also realized that I need to read a lot more proofs relating to integral domain and fields to become confident on the subject.  I tried doing the homework early, and got stumped on a few.


Exciting/Relational)
I see how this crazy notation is useful in describing a field of quotients, but I'm so stumped right now on the concept that I fail entirely to understand or apply the reasoning.  I appreciated the comment that thenotation is useful in helping avoid the assumption that everything works as we are used to it working...because the proof was very tiresome.

(reflect) Due January 27

How long have you spent on the homework assignments?  about 1-1.5 hours on average

Did lecture and the reading prepare you for them?  YES!
What has contributed most to your learning in this class thus far?  Trying to understand everything before coming to class
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 value reading other people's proofs the most, and seeing the approach to writing the proof and the thought process.  I learn best with examples of proofs, and trying to reason why each step was important.  I really appreciate the detailed explanations of proofs in class and reading through extra examples outside of class!!!

Thankyou!

This is my favorite math class thus far!

Focus on Math Extra Credit

A mapping is Surjective if and only if the codomain is equal to the range.

A mapping is Injective if every distinct element of the domain leads to or gives a distinct element in the co-domain.

 A bijective function is a mapping that contains both properties of injective (one-to-one) and surjective (onto).

These properties build to form other identities and tools for graphing in computer programming, and image definitions in computers.  Last semester I was able to write a python program that opened a file of a picture of text, and converted the picture into a character on the computer using the mapping relationship properties and linear algebra to define vectors and matrices.  By taking a linear transformation on the data sets, and isolating the eigenvalues of the matrix which represents the image, I was able to generate code that enabled the image to be captured as text.

I am working on developing an application that will allow you to take a picture of grocery receipts and automatically balance your checkbook.  I plan to convert this application into a tracking measure to also tabulate the rounding error that grocery stores take on individual transactions, and use it to ask for a reimbursement from stores that I frequent over the year!

I see how valuable the properties of injective, and surjective mappings are as a foundation of a lot of different mathematical tools!  I hope to continue to learn about these principles and grow in understanding so that I can continue to explore the possibilities of application of math in my daily life!

3.3 Due January 25

Difficult)
Why is it essential that for theorem 3.12 that for parts 4 and 5 that R ( or the domain of the mapping) is a ring with Identity and that f (the mapping) is surjective?  I know that a ring with identity is a ring that has a multiplicative identity such that 1a=a=a1 for all a in R, and that surjective means that for every element in the co-domain, there exists at least one element in the domain that maps to it.

For us to be using these to show that the mapping of the domain's multiplicative inverse is defined as the multiplicative inverse of the codomain we cannot show this exists without the mapping being surjective...this idea seems really straightforward now that I put it into words, but initially it was stumping me why it was necessary for the mapping to be surjective.

Awe-Inspiring)

It intrigues me how injective and surjective mappings are so essential to mathematical deffinitions.  In differentiation, and rings, and sets, and topology...and probably more, but i've used them in all of those classes fo far.  It makes me think that would be a good research field.  It seems these two things are diverse and all encompassing...the only distinction between isomorphism and homeomorphism.

I took and ACME class that pretended to teach these things, but they make so much more sense in the context of rings as an introduction that they do as a brief section in a tolology chapter.  I really enjoy the way this book introduces an idea, and then repeats it in different examples and un-examples.  This may very well be my favorite math book.

3.2 Due January 22

Difficult)
I'm struggling to see the value and usefulness in not having a zero divisor, or similarly in having an integral domain.  I need to review and re-apply the understanding of the integral domain so that I understand it's application better.
After reviewing a lot of different websites, I'm still stumped...I was getting caugth up on the definition of the O and 1...no not binary, but the additive identity O and the multiplicative identity 1
The integral domain is dependant on the existance of both and that they are not the same.  There are some rings for which the additive and the multiplicative are the same.  The integral domain is a distinction that forces these to be diferent, or that 1 does not equal 0.


Inspiring/Relational idea)
It looks like a unit is a field that doesn't commute. but that there is also a unique solution to au=1 instead of simply having a solution for any a in the ring.  The unique solution is a very powerful tool in defining the tool if subtraction,  Very useful way to manipulate rings that are not commutitive!!!

3.1 (part 2) Due January 20

Difficult)
Theorem 3.1 along with the example preceding it are a new concept to me.  The context of a field or a ring made out of a cartesian product is confusing to me and I need a lot more explanation for it to make sense...
Are we doing the cross product?  FROM WIKIPEDIA I copied the definition.  Would have been nice in the book.
That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Interesting or question about the reading)
I am curious.  The reals are a ring, and also a field, and the quotient numbers are a subring of the reals, and also a subfield.  Where the integers are a subring of the quotient numbers, proving a subring is actually easier than proving the superset is a ring.  We only need to show closure under addition and multiplication, the additive identity, and the additive inverse.

3.1 Due January 15

(Difficult)
The abstract idea of a ring is going to take some getting used to...
The 1R (ring with identity) was a difficult concept to understand and the examples were very helpful.  I also needed the un-example, and I think a few more un-examples would be a helpful addition to the book.  I searched around to try to get a better understanding.

Interesting)
Modular arithmetic and equivalent classes are a very powerful lead-in to the ring identities.  I tried reading about rings before semester started and was confused, but after reminding myself about equivalence classes, and primality I see that it was a great way to set up for introducing the concept of a ring in the definition of systems in mathematics.

2.3 Due January 13

Difficult concept:
theorem 2.8 had a challenging proof concept map.  I spent 15 minutes working through the proof in order to catch onto the steps in the proof.  I really needed the illustration techniques explained in order to better write and formulate a proof.


Reflective:
the set of all congruence classes modulo p (where p is prime) is a powerful statement and I can see a lot of application for applying this to number theory.
I spent some time to memorize the theorem 2.8
let p > 0 then p is prime => for each a!=0 in Zp, ax = 1 has a solution in Zp =>
Whenever ab = 0 in Zp then a=0 or b=0 => that p is prime.
it can also be states that each of the three points of the theorem are equivalent for  p > 0.

2.2 Due January 11

I think the idea of addition and multiplication on congruency sets is fascinating and powerful, but I would like to see some more examples on operations and verify for myself how this works and if it defines a metric space or has some relation to a specific metric space.

I enjoyed the table and the application in hands-on application and verification of sets.  I look forward to working on the problem sets and challenging my understanding on the reading to show myself what I missed.

(I forgot about the blog and quickly did a skim reading to to the blog.  I'll update it after working on the problem set for the homework.)

2.1 Due January 8

I am curious if these concepts apply in the reals or complex numbers and not only in the integers?

Question answered.  Only on integers is modular arithmetic defined.

The definition of [a]=[b] was a difficult concept to pick up and I appreciated the examples in the book that explained how the definition applies for the congruency relationship of a congruent to b mod n.

I enjoyed writing out and solving the proof using symmetry, transitivity, and reflexivity.  I also liked the example of proving that a-b=nk, b-c=ln, then a-c=mn.  But I still want to check and prove if k=l=m...?

1.1-1.3 Due on January 6

The most difficult concept in this reading section is the understanding of the definitions of the divisor, and the application points of prime numbers.  I had to do a few simple example problems to convince myself of the fact of the Fundamental theorem of Arithmetic to verify that the statement was true.  It helped to notice that the definition of p being prime is true if and only if -p is also prime!

I really enjoy the Euclidian algorithm, and using it to find (a,b) or the greatest common divisor.  It is also very useful in knowing and applying understanding for prime numbers.  I'm excited to use these concept to describe modular arithmetic and building on these principles.

Introduction Due on January 6

Introduction Due on January 6

What is your year in school and major?
I am a junior majoring in Mathematics

Which post-calculus math courses have you taken?  (Use names or BYU course numbers.)
112,113,313,314,334,290,341,344,345,320,321(acme junior year first semester)

Why are you taking this class?  (Be specific.)
Requirement for graduation.  and to become more proficient in analysis proofs

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?
Least effective was Dr. Bell--I stopped attending and taught myself.  Didn't have structured lesson plans and was hard to follow his train of thought.
Most effective was Dr. Jarvis--Helped clarify small details and made complex theorems clear and concise.

Write something interesting or unique about yourself.
I am a father of 4 children, and studying to become a seminary teacher.  I ahve had two internships as an object-oriented programmer, and have another this coming summer with Intermountain Healthcare.

If you are unable to come to my scheduled office hours or the TA's scheduled office hours, what times would work for you?
I have cs 235 at the same time that office hours are held.  I can attend office hours anytime Tuesday/Thursday, or MWF: before class any time, or after 4pm.