Paul Pollack

• Homework #1due 1/24PDF
• Handout on properties of the integersPDF
• Syllabus (static copy)PDF

• 1/17Even and odd numbers. Division Algorithm in $\mathbb{Z}$. Divisibility (def + basic properties). Statement of Euclid's algorithm.
• 1/15Proof that $1$ is the least element of $\mathbb{Z}^{+}$. Principle of Mathematical Induction. Exponents. Binomial theorem.
• 1/13Proof $a\cdot 0 = 0$. Laws of inequalities. $1 \in \mathbb{Z}^{+}$. Statement that $1$ is the least element of $\mathbb{Z}^{+}$.
• 1/10(S)no(w) class.
• 1/8Simple consequences of our axioms: Uniqueness of identities. Uniqueness of additive inverses. $(-1)a=-a$. $0\cdot a=0$.
• 1/6Review of syllabus. Fundamental properties of $\mathbb{Z}$.

At this point in your mathematical career, you have accumulated a wealth of experience computing with integers, real numbers, and complex numbers. These items are so familiar to you that they may even have the appearance of being God-given --- as if understanding those objects is What Mathematics is All About.

Abstract algebra challenges this notion. Integers, real numbers, and complex numbers are indeed fantastic, but they are fantastic not because they are handed down from on high, but because they have a rich and theory with useful consequences. And given how useful these objects are, we are compelled to isolate (and consider in the abstract) their most important properties. Once we do so, we find that there are many other objects with the same sorts of interesting properties. For example, we will see that the integers are an example of what is called a ring, that the real numbers and complex numbers are fields, and that the nonzero real numbers form a group. These more abstract-seeming objects are not just interesting in an intellectual let's-talk-about-this-over-coffee kind of way, but understanding them deeply often leads to a new understanding of the objects of original interest.

Let me try to bring this back down to earth: Some primes, like 5, can be written as a sum of two squares: $5 = 1^2 + 2^2$. And other primes, like 3, cannot. The question of when this is possible is a question about the integers. But the easiest way to answer this question is to visit a totally different mathematically system, the ring of Gaussian integers. MATH 6000 students can expect problems about this!

(The official UGA course description, including learning outcomes, can be found here.)

• 

  Abstract Algebra: A Geometric Approach

  by Theodore Shifrin

  Errata Purchase

We will aim to cover Chapters 1--4 and parts of Chapter 6. For further adventures in Algebra-land, I encourage you to take MATH 4010/6010.

There will be three fifty-minute in-class exams, as well as a final exam.

• Midterm #1: Friday, February 14
• Midterm #2: Friday, March 28
• Midterm #3: Friday, April 18
• Final exam: Wednesday, April 30 12:00 - 3 PM (location TBA)

No make-up exams will be given. The final exam is cumulative. Your grade is made up of the following weighted components:

• Each midterm: 15% (total of 45%)
• Homework: 25%
• Final exam: 30%

This class falls into the interactive lecture genre (not entirely unrelated to the practice of call and response in a liturgical context). What this means is that I intend to punctuate the lectures frequently with questions for you. For the show to go on, class participation is absolutely essential. Since you cannot participate in class if you are not present in class, your attendance is required. In particular, more than four unexcused absences may result in you being automagically withdrawn from the class. Of course, missing class is sometimes a necessity; keep me posted whenever you have a conflict and we should not have any issues.

Homework will be collected roughly once each week. As a general rule, late homework assignments are not accepted. Your lowest HW score will be dropped at the end of the semester.

All exams are closed book and closed notes.

Students enrolled in MATH 6000 will take the same exams as the students in MATH 4000 but will be assigned additional homework problems.

You are not only allowed, but encouraged to collaborate with your classmates on the homework assignments. The joy of mathematical discovery was meant to be shared! Having said that, collaboration does not mean copying, and consulting AI tools such as ChatGPT is not "collaboration" in the sense intended. You may not copy solutions from a textbook, classmate, website (including an AI tool), etcetera, and you must be the one to handwrite (or type) your solutions.

By entering UGA, you have already agreed to abide by the UGA honor code: "I will be academically honest in all of my academic work and will not tolerate academic dishonesty of others." A Culture of Honesty, the University's policy and procedures for handling cases of suspected dishonesty, can be found at honesty.uga.edu. A good rule of thumb is that you should be able to explain any work you turn in to a hypothetical interrogator.

Students with disabilities who may require special accommodations should talk to me as soon as possible. Appropriate documentation concerning disabilities may be required. For further information, please visit the Disability Resource Center page.

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Additional information, including free digital well-being resources, can be accessed through the UGA app or by visiting https://well-being.uga.edu.

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