Introduction to Abstract Algebra

Fall 2016

For most students, this course will be the first serious venture into the world of abstract mathematics. Similar to the jump from a specific real number to a variable, we will, in a certain sense, treat the operations '+' and '·' as variables. We will see what the essential properties of the operations are and how varying these properties results in different algebraic objects.

Another measure of the level of abstraction is that the integers modulo n will serve as examples of groups, rings, and fields (these are the objects we will study). Among other examples, we will look at the group of matrices under matrix multiplication and permutations of finite sets (where multiplication corresponds to composition).

There are numerous applications of abstract algebraic objects in mathematics and physics; our course, however, will concentrate on the abstract side. There will be many opportunities to see how rigorous reasoning actually leads to discoveries (while in lower-level courses, rigorous reasoning is often a way to confirm a discovery made through other means). Developing and increasing the capacity for abstract thinking is an important long-term objective of this course.

More often than not, mathematics is a collaborative activity. Mathematicians (both in academia and in the industry) rely on their peers to review their work to see if there are any gaps in the arguments. For you, peer review is also an efficient way to learn proofs. That is why this course will emphasize team activities.

This course will use a different correspondence between the numeric grades and the letter grades than the more usual 90-80-70 scheme. The percentage score 80% is the dividing line between A and A-; 72% and above is a B; and 60% is (barely) a C. For written homeworks, this means that the score of 20 of 25 is an A; 18 of 25 is a B; and please try not to get below 15 too often. On the exams, this means that you can skip a problem and still get an A, if all the other problems are solved correctly.

This grading scheme takes into account the level of difficulty of the assignments and the exams as well as the rigorous grading style of your instructor.

Ideally, you should work with your team to solve each homework problem absolutely correctly and present a neatly written (better, typeset) solution by the time the homework is due. This will guarantee the prefect written homework score as well as the full 10 point credit on the "homework component" for exams 1, 2, and the final. The rest of this section explains what happens if your team did not manage to solve some of the homework problems correctly.

First, your homework score (the one that's worth 20% of your final grade) is the average of the grades you get for each problem when you submit the work for the first time. In addition, you will lose up to 10 points on both mid-term exams, as well as on the final, unless you revise (i.e., solve correctly by the revision cut-off date) each of the homework problems that received a bad score. The revisions have to be done individually; each revision should be a self-contained solution of the problem.

The grading scheme for homework problems is explained in the Homework Guide. To get the full 10 points on the exam, you need to have a grade 4 or higher for each of the homework problems. To get 5 points, you need to have a grade of 3 or higher for each of the problems. If there is a problem with the grade 2 or lower, the homework component grade will be 0.

Revisions can be submitted any time, in person or electronically, before the start of the corresponding exam. There is one restriction (meant to discourage procrastination): you can submit only three revised solutions in any 24 hour period.

If your presentations and participation/quizzes are consistently good (feel free to ask me for feedback if you are not sure), I will add a third of a letter grade to your final grade. At worst (if you are not ready or miss the class half the time), I will subtract one full letter grade from your final grade.

You are welcome and encouraged to ask me for help with course material during the office hours or by making an appointment. The rule for team assignments is that as least two of the team's members must be present.

You should expect to spend a considerable amount of time working on this course outside of class. According to the US Department of Education definition, a credit hour is ''an amount of work that reasonably approximates not less than one hour of classroom ... instruction and a minimum of two hours of out of class student work each week for approximately fifteen weeks''. This is a 4-credit hour course, so you need to budget 8 hours per week for independent work on this course. At the same time, try to be efficient. If you are spending a lot of time on homework without making much progress, talk to me.

Participation in homework teams is an essential part of the course. If you miss a team meeting, it is your responsibility to work out what form should your contribution to the team effort be. Putting your name on a team homework if you have not participated in meetings is a violation of University's academic integrity policy.

The use of external to the course sources of solutions to homework problems is allowed, provided the source is fully cited and the material is critically evaluated. If you do not cite a source for the information, or copy a solution without a credible attempt to understand it, then you are violating the University's Academic Integrity Policy.

Tests and quizzes are to be written strictly individually. University's Academic Integrity Policy will be enforced; everyone caught cheating on any assignment will face a range of penalties, up to a failing grade in the course.

Normally, no make-up tests will be given; if for some reason you are not able to take a test, please, let me know as soon as possible.

If you have questions or concerns about the course, be sure to discuss them with me during my office hours or by appointment.

Everyone participating in this course is expected to be respectful of each other without regard to race, class, linguistic background, religion, political beliefs, sex, gender identity or expression, sexual orientation, ethnicity, age, veteran's status, or physical ability. If you feel these expectations have not been met, please feel free to discuss it with me or with the designated diversity liaison Dr. Elizabeth Goode.