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.

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