Review for final, Due Wednesday, April 11

• Which topics and theorems do you think are important out of those we have studied?
• I think that the most important topics that we have studied in the last 3rd of the class have been classifing groups, Lagrange's theorem, the isomorphism theorems for groups and Cauchy's theorem
• 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.
• I need to go through and learn all of the proofs and memorize theorems and definitions
• How do you think the things you learned in this course might be useful to you in the future?
• I don't know how this will be useful in the future. Hopefully it will help me in my upper division courses next year, Number Theory, Cryptography and Combinatorics.

Section 9.4, Due Monday, April 9

The section for today was 9.4: The field of Quotients for an Integral Domain.

This section wasn't that difficult to understand. The hardest part was probably the proof of Lemma 9.26

I thought that this was an interesting argument for the construction of Q the rational numbers.

Sections 8.4, 8.5, Due Friday, April 6

The sections for today were 8.4: Conjugacy and the Proof of the Sylow Theorems, and 8.5: The Structure of Finite Groups.

The most difficult part of these sections was the proofs of th Sylow Theorems.

I thought that the end of 8.5 was interesting when they had a complete classification of all groups of order =< 15.

Section 8.3, Due Wednesday, April 4T

The section for today was 8.3: The Sylow Theorems.

This section wasn't that difficult...maybe because there weren't very many proofs. But the proof of corollary 8.18 was kind of hard to understand.

I thought that this section was pretty interesting. It is neat that we will be able to classify non-abelian groups along with abelian ones.

Section 8.1, Due Monday, April 2

The section for today was 8.2: Finite Abelian Groups.

This section was very complicated. The hardest part to follow was the proof of Lemma 8.6.

I thought that Lemma 8.8 was interesting. We used on the very first test, and now it's proof is understandable because we have learned so much since then.

Section 8.1, Due Friday, March 30

The section for today was 8.1: Direct Products.

The most difficult part for me was in the proof of Theorem 8.1 when they showed that f was a homomorphism.

I thought that it was an interesting section. The fact that you can create an internal direct product, made up of normal subgroups, that is isomorphic to the original group, is pretty neat.

Section 7.10, Due Wednesday, March 28

The section for today was 7.10: The Simplicity of An.

This section was pretty abstract. I thought that the proof of Theorem 7.52 was the most difficult to understand.

I think that the fact that was revealed at the end was the most interesting. It stated that If n>=5, then (1), An, and Sn are the only normal subgroups of Sn.

Section 7.9, Due Monday, March 26

The section for today was 7.9: The Symmetric and Alternating Groups.

This section was kind of confusing. I think that most difficult part was the proof of Lemma 7.49. The whole concept about even/odd is pretty abstract.

I thought that the cycles were pretty cool and that the notation was fun and interesting.

Section 7.8, Due Friday, March 23

The section for today was 7.8: Quotient Groups and Homomorphisms.

I thought that the most difficult part of this section was the proof of theorem 7.44.

I thought the end was cool when they talked about Group Theory research. I can't believe that that proof was 10,000 pages.

Section 7.7, Due Wednesday, March 21

The section for today was 7.7: Quotient Groups.

The most difficult part of this section was the proof of theorem 7.36.

I thought that the most interesting thing about this section was theorem 7.37.

Section 7.6 (second half), Due Monday, March 19

The section for today was 7.6: Normal Subgroups.

The most difficult part in this section was Theorem 7.34 and its proof.

The most interesting part was 7.33 which uses normal subgroups to convert theorem 6.5 to groups.

Section 7.6 (first half), Due Friday, March 16

The section for today was 7.6: Normal Subgroups.

The most difficult part of this section so far was the fact that aN=Na does not imply that an=na for all n.

I thought that this was an interesting parrallel between rings and groups. I guess normal subgroups are the equivalent (kind of) to Ideals in rings.

Section 7.5 (second half), Due Wednesday, March 14

The section for today was 7.5: Congruence and Lagrange's Theorem.

I thought that the most difficult part of this section was Theorem 7.30; its proof is very complicated.

I thought that it was interesting that mathematicians are trying to to classify all finitie groups up to isomorphism. That sounds like a really massive project. It would be really handy if they were able to do that.

Section 7.4 (first half), Due Monday, March 12

The section for today was 7.5: Congruence and Lagrange's Theorem.

The most difficult part for me was Lagrange's Theorem.

I think that it is neat that congruence also transfers from rings to groups, with modification of course.

Section 7.4, Due Friday, March 9

Today's section was 7.4: Isomorphism and Homomorphisms.

The most difficult part of today's reading was understanding what it means to have an automorphism or an inner automorphism.

I thought it was cool at the end when they said that by the use of representations the study of group theory can be reduced to the study of permutation groups. That is a really convenient fact.

Midterm 2 Review

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

A: I think that the most important topics/theorems are the ideals, the first isomorphism theorem, criteria for groups and cyclic groups.

Q: What questions do you expect to see on the exam?

A: I expect to see questions like those on the study guide. Proofs, examples etc.

Q: What do you need to work on understanding better before the exam?

A: I need to work on finding examples of different things.

A problem I would like to see worked out in class is Give an example of a maximal ideal in a ring that does not contain all proper ideals of the ring.

and #7 on the review sheet.

Section 7.3, Due Monday, March 5

The section for today was 7.3: Subgroups.

The most difficult part of this section was understanding Generators of Groups.

I think that this section is interesting. Cyclic Groups seem very neat, I liked reading about them.

Section 7.2, Due Friday, March 2

The section for today was 7.2: Basic Properties of Groups.

I did not understand Corollary 7.9. Hopefully we go over it in lecture.

I thought that it was interesting that groups have these properties. I liked the example you did last time where we were able to find all the elements of D4 by multipling random ones together.

Section 7.1 (part 2), Due Wednesday, February 28

The section for today was the second half of 7.1.

I thought the most difficult thing about this section was rotation groups.

I like that rings can be made into groups. I thought that it was interesting that every ring was an abelian group under addition.

Section 7.1, Due Monday, February 27

The section for today was 7.1: Definition and Examples of Groups.

The most difficult thing in this section was finding inverse permutation funtions to get the identity.

Judging on what I have read so far, groups seems like they will be interesting to learn about. I am curious to see how similar/different they are compared to rings.

Section 6.3, Due Friday, February 24

The section for today was 6.3: The Structure of R/I When I is Prime or Maximal.

The most difficult thing about this section was idea of maximal rings.

I would just like to reflect on rings in general seeing as this is the last section about rings. I think that they are very interesting and have a lot of convenient properties. It's cool that the same properties hold for the very general cases.

Section 6.2 (Part 2), Due Wednesday, February 22

The section for today was the second part of 6.2: Quotient Rings and Homomorphisms.

The most confusing/difficult part of this section was the proof of theorem 6.13, it was hard to follow.

I thought that this section was interesting. I am wondering how this use of the term kernal relates to the kernals I learned about in Linear Algebra and Multivariable Calculus. I guess they seem similar.

Section 6.1 (part2) and section 6.2 (part 1). Due Tuesday, February 21

The reading for today was the rest of section 6.1: Ideals and Congruence, and then the first half of section 6.2: Quotient Rings and Homomorphisms.

The most difficult concept for me was the idea of "cosets."

I am wondering how general these ideas can possibly get. It seems that each section just generalizes further than the last, making it more "abstract." I guess that is the point. I believe that we are near the end though. The next chapter is about groups, but I am sure we will be using some of the ideas from the first six chapters.

Section 6.1 (part 1), Due Friday, February 17

The section for today was the first half of 6.1: Ideals and Congruence.

I thought that the most difficult thing to understand about ideals is the distinction between principal ideals and non-principal ideals.

I thought that the concept of ideals was very intriguing. The absorption property was very interesting--a ring that behaves ideally.

Section 5.3, Due Wednesday, February 15

The section for today was 5.3: The Structure of F[x]/(p(x)) When p(x) is irreducible.

The hardest part to understand was theorem 5.11, the fact that the extension field contains a root to p(x).

I thought that it was fascinating that the definition for the complex numbers comes from these ideas and structures.

Section 5.2, Due Monday, February 13

The section for today was 5.2: Congruence-Class Arithmetic.

The most difficult part of this section was Theorem 5.9. I don't see how a non-constant polynomial can be a unit.

I think that it is great that addition and multiplication, along with other theorems transfer to F[x] so nicely.

Section 5.1, Due Friday, February 10

The section for today was 5.1: Congruence in F[x] and Congruence-Class Arithmetic.

The most difficult part of this section, for me, was the concept of infinitely many congruence classes for R[x]/(x^2+1).

I think that it is really interesting/convenient that the ideas and rules about congruence classes transfer so nicely to the polynomial rings.

Sections 4.5 & 4.6, Due Wednesday, Feb 8

The sections for today were 4.5: Irreducibility in Q[x] and 4.6: Irreducibility in R[x] and C[x].

The most difficult thing in these two sections, for me, was Einstein's Criterion (Theorem 4.23) and its proof.

I was very interested to learn the proof of the Rational Root Theorem. I have used it a lot in various math classes and while tutoring in the math lab. It is nice to learn where it comes from.

Questions about Reviewing for the test:

Which topics and theorems do you think are most important out of those we have studied? The Division Algorithm, Rings and their properties, isomorphisms and homomorphisms

What kinds of questions do you expect to see on the exam? I expect to see problems asking if sets are rings, if two rings are isomorphic or homomorphic to eachother, proofs about divisiblity, factoring and dividing polynomials, functions induced by polynomials, etc.

What do you need to work on understanding better before the exam? I need to read through my notes and study the proofs listed on the review sheet. I also should go over the properties of rings, fields, and integral domains.

I would like to see this problem worked out: If there exists a ring isomorphism A

B we write A = B. Prove that = is an equivalence relation

on the class of all rings. .

Section 4.4, Due Monday, February 6

The section for today was 4.4: Polynomial Functions, Roots and Reducibility.

The most difficult thing about this section was the Proof by Induction of Corrollary 4.16.

I thought it was very interesting to learn about the distinction between a polynomial as an indeterminate or transcendental element and as a function induced by a polynomial element.

Section 4.3, due Friday, February 3

The section for today was 4.3: Irreducibles and Unique factorization.

I think that the most difficult part of this section is the idea of "associates."

Again, I think that it is very interesting that these properties of irreducibles (primes) carry over from the integers to polynomials.

Section 4.2, Due Wednesday, February 1

The section for today was 4.2: Divisibility in F[x].

I thought that the most difficult part of this section was the proof of Theorem 4.5 which claims a unique gcd of any two functions f(x), g(x).

I thought that it was very interesting and convenient that the properties of divisibilty practically transfer straight over to Polynomials. Most of the proofs are exactly the same, except that for the monic condition on the gcd.

Section 4.1, Due Monday, January 30

The section for today was 4.1: Polynomial Arithmetic and the Division Algorithm.

I found the proof of the Division Algorithm for polynomials to be the most difficult part of the reading. I'd appreciate going over it in the lecture.

I thought that it was interesting that polynomials can be examined in terms of rings. I'm amazed at how interconnected math can be.

Due Friday, January 27

I spend about one to two hours on each assignment. I think that the reading and lectures are generally good at preparing me for the homework, but sometimes there is a disconnect between what I hear in class/read in the textbook and what I am asked to do in the homework. I think that the lectures should bridge that gap instead of just repeating what is in the chapter. Reading the textbook and then soldifying that information in lecture has been the most beneficial to my learning thus far. As I said before, I think lectures should address the coming homework more. But I do like what you are doing already (going over the section & the proofs), but I would appreciate a little heads up for the homework too.

3.3, Due Wednesday, January 25

The section for today was section 3.3: Isomorphisms and Homomorphisms.

I thought that Theorem 3.12 was pretty confusing. I wasn't able to follow the proofs very well. Hopefully they will make sense tomorrow in class.

I thought that the concept of isomorphisms and homomorphisms was very interesting. I particularly liked the analogy that they drew to Roman Numerals. I took Math 300, The History and Philosophy of Math, and we explored a lot of different numeral systems. I mean any different numeral system, or more abstractly, language systems are isomorphisms, very interesting stuff.

3.2, Due Monday, January 23

The section for today was 3.2: Basic Properties of Rings.

I thought that the concept of "units" was kind of confusing. It would be helpful if you went over that in class.

I thought that the reading was pretty interesting. I liked the introduction, which reminded the readers that far more properties are used in Z than just the ring axioms. So naturally, there will be more properties that apply to rings in general.

The rest of 3.1, due Friday, January 20

The section for today was the second half of 3.1, Definition and Examples of Rings.

I didn't think that this was particularly difficult, but if I had to choose something, I'd say theorem 3.1. I'd like to see the proof of that.

I thought that it was an interesting section. I liked learning about subrings. I liked the last example; it was so simple and logical.

3.1 (first half), due Wednesday, January 18

The reading for today was the first half (through page 48) of section 3.1.

The most difficult part for me in this section was the part with the matrices and why they are rings. But upon carefully reading it, I was able to understand.

This was a very interesting section. I particularly liked the introduction to chapter 3. I was very interested to learn why this discipline is called abstract algebra. I thought that it was just because it was going to be very difficult. But, according to the book, it is called abstract because the "common core of essential features" is "abstracted" from different systems so that they can be more easily compared/examined. Very interesting!

2.3, due on Friday, January 13

The section for today was 2.3: The Structure of Zp when p is prime.

I found Theorem 2.8 kind of tricky, I'd like to see that proof in class.

I thought it was very interesting that Zp when p is prime would have these certain properties, they are so nice and convenient. I found the proofs pretty straight forward and logical. I can't wait to learn the implications of these properties. It started to sound a lot like linear algebra in this section, so hopefully I will be able to draw some connections between the classes.

2.2, due Wednesday, January 11

The reading for today was section 2.2: Modular Arithmetic.

What I found difficult about the reading was probably the transfer of the regular arithmetic and multiplication properties to modular arithmetic and multiplication. But that isn't so hard, it's just probably the hardest in the section.

I thought that the material was pretty interesting, especiallly the simplicity and logic of the modular form. I hope that we learn more about applications of modular arithmetic and multiplication in class; I know that there are plenty fascinating real-world uses of modular form.

2.1, Due Monday, January 9

The reading for today was Section 2.1--Congruence and Congruence classes.

What I found difficult about this section was the last proof of Corollary 2.5, part (2). I think I found this difficult because it was kind of long and hard to follow, but I thought about it some more and it makes more sense now.

I haven't thought about congruence and "mod n" for a very long time. I remember learning about it 290, but that was two years ago. I do remember however, in the Seminar in Mathematics that I took last semester that there was a presentation in which the speaker talked about some of the applications of congruence (like license plates). It was very interesting, hopefully it won't be to difficult to relearn.

1.1-1.3, due Friday, January 6

The most difficult part of the material in sections 1.1-1.3, for me, was the proof of the division algorithm. It was really long and complicated. But I re-read it and understood it. It was beautiful in its logic. I also really liked the logical flow of The Fundamental Theorem of Arithmetic.

This reading was very interesting. I especially liked learning Euclid's algorithm. I've been reading a book entitiled, "The Advent of the Algorithm" by David Berlinski. It is a history of how the idea of algorithms was developed and its implications for our modern world (which is basically built upon algorithms). It is interesting to learn of some of the ancient algorithms (Euclid being from Classical Greece if I'm not mistaken.) As a double major in Math and History, I find the history of math very interesting. In fact, I took Math 300 (The History and Philosophy of Math) last winter and throughly enjoyed it.

Introduction

Welcome to my Abstract Algebra blog!

My name is Lorraine Hilton and I am in my third year of my undergraduate--double majoring in Math and History.

Post-calculus, I have taken 290 (Fundamentals of Mathematics), 313 (Linear Algebra), 314 (Multi-Variable Calculus), 300 (History and Philosophy of Mathematics), 341 (Theory of Analysis), 334 (Differential Equations) and I am currently enrolled in 371 (Abstract Algebra--obviously) and 352 (Complex Analysis).

I am taking this class because it is required to graduate with a math degree. But I am also curious about the subject--I don't really know what to expect.

The most effective math professor I ever had was Professor Villamizar. He was very clear in explaining the material. He was able to explain it at the students' level very effectively. Also, he sincerely cared about the students and brought the spirit into class. He was a genuine person and an excellent professor.

An interesting fact about me is that I really like to run. I have run two marathons and two half-marathons. I am preparing for a third half-marathon in March!

I cannot come to your scheduled office hours 2-2:50 MWF. I could come 3-3:50 MWF or 1-1:50 TTH or 4-4:50 TTH