Eloísa Grifo | Math 412

Math 412: Introduction to Modern Algebra

Winter 2019 Sections 1 and 2

Syllabus

Dr. Eloísa Grifo (please address me as Eloísa)

Course assistant: Pleum Piriyatamwong


Welcome to Math 412!

Math 412 is an introduction to abstract algebra, required for all math majors but possibly of interest also to physicists, computer scientists, and lovers of mathematics. We will begin with ring theory: our first goal is to prove the Fundamental Theorem of Algebra, about the ring you've been studying since elementary school, the integers. In the second half, we will study group theory. In addition to developing many examples, students will prove nearly all statements in this course.

Warning: we differ from the book by including in our definition of ring that every ring contains 1.

Prerequisites: Math 217. Students are expected to know linear algebra and to have had a course in which they have been trained in rigorous proof techniques (induction, proof by contradiction, etc).

Required text: Abstract Algebra: an introduction by Thomas W. Hungerford, 3rd edition (earlier editions are OK but homework numbering and page numbers may differ).

Course expectations: Math 412 students are responsible for learning the material on their own through individual reading of the textbook before coming to class. In class, you will work together on more theoretical concepts in small groups using worksheets; this is an essential part of the course and your grade. The course is run "IBL" style, similar to Math 217, so be prepared every time! You will be expected to work out more computational exercises on your own, which will be tested by weekly webwork. You will also have a Quiz every Tuesday, a graded, written problem (think Math 217 Part B) set due Thursdays, and Webwork due every Friday.
Attendance is required.

Sections: All sections will use the same Syllabus, do the same classwork, have the same webwork, take the same exams, and do the same homework, regardless of instructor.
  • Section 1 meets every Tuesday (East Hall 4096) and Thursday (East Hall B737) from 8:30 am to 9:50 am
  • Section 2 meets every Tuesday (East Hall 3096) and Thursday (East Hall B735) from 10:00 am to 11:20 am
  • Section 3 is taught by Dr. Jack Jeffries




Office hours

Tuesdays 5 — 6 pm
Wednesdays 3 — 4 pm
Thursdays 5 — 6 pm

Webwork

Webwork is due every Friday at 11:59 pm.

Problem Sets

Problem sets are due every Thursday.

Review materials


How to prepare for the final?
  • Know all the important definitions and theorems.
    One great way to do that is to go over all the worksheets and focus on the information inside a box. Write your own lists of definitions and theorems. Make sure you not only remember these definitions and theorems but also that you understand what they mean.
  • Have some basic examples at hand.
    Go over your list of definitions and add examples and non-examples to each particular definition. You might also add some examples of special cases where you would use some of the most important theorems to your list of theorems.
  • Practice some computational examples too.
    Webwork is a great source of lots of computational examples. What groups are isomorphic to each other? Can you apply the Euclidean algorithm? Can you count how many units are there in a particular ring? Webwork will show you the correct answers, if you do not know how to obtain them, ask me for help!
But most importantly, you are very welcome to ask me questions during office hours.

Exams

The Midterm was on Tuesday, February 26, 6 to 8 pm, in CHEM 1400. (Midterm solutions)
Common mistakes on the midterm

The Final Exam was on Thursday, April 25, 4 to 6 pm, in EH 1360. (Final exam solutions)

How to prove it

On the Importance of writing well, a commentary from Ravi Vakil. Everything he says about the importance of writing well applies also to writing your Math 412 homeworks!

Why all rings should have a 1

Supplement on group actions written by Karen Smith

To read before the first class: sections 1.1 and 1.2 in the book.

After Math 412: Here you can find a list of IBL courses offered by the UMich Math Department.

Alternatives to Math 412:
Math 312 also satisfies the algebra requirement for the math major. This course covers much of the same material but demands a bit less in terms of what you are expected to be able to prove. It might be a better option for you if you do not like proofs, struggle with or have not had a good introduction to proofs like Math 217. Math 312 will cover some proof techniques that we will assume in Math 412.
Math 217 If you haven't had this, take it! Math 217 is not just "matrix algebra" — it is more theoretical. It will teach you how to "do proofs" for future math classes. It is a great class, with applications all over science, engineering, and math, and the perfect prereq for Math 412.
Math 490 This is a different topic (topology) but also moves at a similar pace and is taught in a similar way. If you are looking for an upper level math elective and don't need an "algebra" course, this is another option.
Math 493 satisfies the algebra requirement, and is also an introduction to Abstract Algebra but it assumes students have had a much deeper introduction to abstract mathematics, such as Math 295

Department of Mathematics | University of Michigan | East Hall | 530 Church Street | Ann Arbor, MI 48109