Course Syllabus

General Information

Prerequisite

Math 24 and Math 71 (students with Math 22 or with Math 31 may enroll with instructor permission).

 

Course Catalog Description

This course provides a foundation in core areas in the theory of rings and fields. Specifically, it provides an introduction to commutative ring theory with a particular emphasis on polynomial rings and their applications to unique factorization and to finite and algebraic extensions of fields. The study of fields continues with an introduction to Galois Theory, including the fundamental theorem of Galois Theory and numerous applications.

 

Textbooks

Official textbook: David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd edition. We will cover parts of Chapter 9 and all of Chapters 13 and 14.

 

 

Other references to look at: James S. Milne, Field Theory https://www.jmilne.org/math/CourseNotes/FT.pdf.

Ian Stewart, Galois Theory (4th edition). 

Juliusz Brzezinski, Galois Theory Through Exercises.

Serge Lang, Algebra (3rd edition). (roughly chapters IV-VI)

 

Scheduled Lectures

Instructor	Jack Petok
Class	MWF 12:50 - 1:55
X-hour	Tu 1:20 - 2:10

Instructor

Instructor	Jack Petok
Office Hours	TBD
Email	jack.petok AT dartmouth.edu

COVID 19 Information

You are expected to attend class in person unless you have made alternative arrangements due to illness, medical reasons, or the need to isolate due to COVID-19. For the health and safety of our class community, please: do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home. 

In accordance with current College policy, all members of the Dartmouth community are required to wear a suitable face covering when indoors, regardless of vaccination status. This includes our classroom and other course-related locations, such as labs, studios, and office hours. If you need to take a quick drink during class, please dip your mask briefly for each sip. Eating is never permitted in the classroom. (The only exception to the mask requirement is for students with an approved disability-related accommodation; see below.) If you do not have an accommodation and refuse to comply with masking or other safety protocols, I am obligated to assure that the Covid health and safety standards are followed, and you will be asked to leave the classroom. You remain subject to course attendance policies, and dismissal from class will result in an unexcused absence. If you refuse to comply with masking or other safety protocols, and to ensure the health and safety of our community, I am obligated to report you to the Dean’s office for disciplinary action under Dartmouth’s Standards of Conduct. Additional COVID-19 protocols may emerge. Pay attention to emails from the senior administrators at the College.

 

Exams

There will be one takehome midterm exam during an entire week at some point in the term.

There will be a final exam on Friday, March 11 at 11:30 am (location TBA).

Homework 

The only way to really figure this stuff out is to work out some exercises on your own. In a typical week, you will be assigned around 6-10 homework problems, announced on Canvas. Homework will typically be released each Wednesday and due the following Wednesday at the beginning of class. If you are experiencing an illness, I will accept a high-quality scan by email. I will not accept any late homeworks, but I will drop your lowest score.

Collaboration is an important part of learning and doing mathematics. You are encouraged to discuss these problems amongst each other, but the final write-up must be your own. You are allowed to use textbooks and notes from class, and you are allowed to use other online reference and educational sources such as Wikipedia, but you are not allowed to specifically look up solutions of the homework problems as a means to avoid thinking about the problem yourself . You are also not allowed to ask for the solution by posting a particular problem on any online Q & A site or help forum.

The homework problems will be a mix of proofs and computations. For full credit, explain all work and write your proofs in full English sentences. I recommend that you type your solutions in LaTeX, but I will also accept neat written work.

 

Expectations

I expect you to attend every class. It is your responsibility to keep informed of any announcements, syllabus adjustments, or policy changes made during scheduled class time. Not all announcements will be posted on Canvas, though I will try my best.

I borrowed this next blurb from my PhD advisor: "In my experience as a student, most people do not follow all the details of a lecture in real time. When you go to a math lecture, you should expect to witness the big picture of what's going on. You should pay attention to the lecturer's advice on what is important and what isn't. A lecturer spends a long time thinking on how to deliver a presentation of an immense amount of material; they do not expect you to follow every step, but they do expect you to go home and fill in the gaps in your understanding. Not attending lecture really hurts your chances at a deep understanding of the material."

Do not fall behind. It is completely normal to not understand the lecture or not be able to solve a homework problem on your own. Attend lectures and try to follow the big picture even if you are get lost in the details. Seek help whenever you need it. There are many resources available to ensure that you are keeping up with the course. I encourage you to ask for help from your classmates and to come to my office hours whenever you are having trouble understanding the course material. I firmly believe that there is no such thing as a stupid question. 

 

The Academic Honor Principle

Academic integrity is at the core of our mission as mathematicians and educators, and we take it very seriously. We also believe in working and learning together.

Collaboration on homework is permitted and encouraged, but obviously it is a violation of the honor code for someone to provide the answers for you.

On written homework, you are encouraged to work together, and you may get help from others, but you must write up the answers yourself. If you are part of a group of students that produces an answer to a problem, you cannot then copy that group answer. You must write up the answer individually, in your own words. A good practice is to discuss ideas on a blackboard, then erase the blackboard and try to reproduce the arguments later, on your own paper, and without assistance. Permitted resources for homeworks include textbooks and notes from class, and you are allowed to use other online reference and educational sources such as Wikipedia, but you are not allowed to specifically look up solutions of the homework problems as a means to avoid thinking about the problem yourself . You are also not allowed to ask for the solution by posting a particular problem on any online Q \& A site or help forum.

On the final exam, you may not give or receive help from anyone. The final exam in this course is closed book, and no notes, calculators, or other electronic devices are permitted. 

On the midterm exam, you will be permitted access to your notes and textbooks, but no other resources allowed unless specified by instructor.

Plagiarism, collusion, or other violations of the Academic Honor Principle will be referred to the Committee on Standards.

 

Grades

The course grade will be as follows:

Written homework	45%
Takehome midterm	20%
Final exam	35%

 

Other Considerations

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with your instructor before the end of the second week of the term to discuss appropriate accommodations.

Students who need academic adjustments or alternate accommodations for this course are encouraged to see their instructor privately as early in the term as possible. Students requiring disability-related academic adjustments and services must consult the Student Accessibility Services office (Carson Suite 125, 646-9900, Student.Accessibility.Services@Dartmouth.edu). Once SAS has authorized services, students must show the originally signed SAS Services and Consent Form and/or a letter on SAS letterhead to their professor. As a first step, if students have questions about whether they qualify to receive academic adjustments and services, they should contact the SAS office. All inquiries and discussions will remain confidential.