Review sheet covering material not includeed in
previous review sheets.
Past test review sheets
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II.
This course will emphasize the art of abstract reasoning in exploring the fundamental objects of Modern Algebra: groups, rings and fields. The notion of group is the mathematical essence of the study of symmetry, which deals with transformations of an object that appear to leave it unchanged (here are some examples of symmetric patterns left invariant by translations and rotations). Much of higher mathematics and some physics depends on this fundamental concept. The notion of ring captures some of the essential algebraic features that integers, matrices and polynomials share. Some of the most sophisticated codes which secure trillions of dollars of banking transactions arise from ring theory.
To get the most out of this course you should read the sections of the text before coming to class. I plan to concentrate on the more challenging concepts without repeating everything done in the book. Reading the text and doing homework exercises are both critical to your success in this course. Hungerford's text is a well written introduction to Modern Algebra. It is delightful to read and full of insight. Unfortunately, it is fairly expensive; perhaps cheaper copies may be found on the web (see the article).
If this is your first "proofs" class, please be prepared to put in a bit of extra effort to learn the language of abstract mathematics.
Course Goals: Our goal will be to cover most of what the text refers to as the "core course", Chapters 1 through 7 and, if there is time, the first two sections of Chapter 8. We will begin by reviewing the material in the first two appendices during the first week. An appealing feature of this book is that it begins with the most concrete part of abstract algebra first (modular arithmetic and polynomials) before dealing with the more abstract notions of rings and groups. See the tentative (ambitious) schedule.
Grading policy: Your basic grade will be a function of your score on the homework, the two tests and the final. It will be computed as follows:
| Tests | 40% |
| Homework | 30% |
| Final Exam | 30% |
Your homework may be substituted for your lowest test score in computing your prefinal average; moreover, if your score on the final exam is higher than your prefinal average, your final exam will be weighted 50% and the other weights will decrease accordingly. The grading scale will roughly be as follows (plusses and minusses may be given in borderline cases):
Homework: Doing Mathematics is the best way of learning it, especially in a class like this; for the skill of writing proofs well only comes through practice. Please feel free to discuss homework problems with me or others. Work will be due on Tuesdays at the beginning of class. Solutions will be posted, so late homework will not be accepted.
To receive full credit, your solutions must be correct, complete and neatly written. Proofs should be written in complete sentences.
The Department of Mathematics and Statistics supports providing equal access for students with disabilities. I encourage any student needing to request accommodations for a specific disability to please meet with me at your earliest convenience to ensure timely and appropriate accommodations.