Text: Contemporary Abstract Algebra by Joseph Gallian
Class: TTh 10:00-11:15, E408
Introduction: Abstract Algebra is the branch of pure mathematics that tries to reduce math to symbolic manipulation. Here is an example. Consider a cube, an object most people feel they understand. 6 congruent square faces, 12 congruent edges, 8 vertices. How much symmetry does it have? This question seems to make sense already, but it is hard to quantify until we define ''symmetry''. In algebra, a symmetry of a geometric object is a bijection of the object to itself that preserves the object's defining properties. ''Preserving the object's defining properties'' in this case means preserving the cube's shape (the vertex-edge-face structure) and its orientation in space. So, how many geometric symmetries does the cube have? Our position in this class is that we don't really understand the cube if we cannot answer this question. Our collective mind will fester around this problem, until we have built up the machinery necessary to answer it. We will see that not only can we count the symmetries, we can describe the structure of this set, using the algebraic notion of group. We will name the symmetry group of the cube.
For the last few hundred years, algebra has developed mostly around attempts to solve specific math problems, mostly in number theory and geometry. For example, algebra was used to solve the ancient problem of determining which polygons can be constructed using only a straight edge and compass. Gauss solved this problem in 1796, when he was five years old, I think. We'll try to solve some of these types of problems.
''Abstract'' means, ''disassociated from any specific instance''. In the 20th century, certain basic algebraic objects were isolated and abstracted to form the core of abstract algebra: groups, rings, fields, and vector spaces. These abstract objects appear in practically every area of pure mathematics. We'll start with groups, then do rings and fields. You can do vector spaces in Math 321.
Prerequisites: Math 221, 250.
Requirements: You must attend the lectures, do the homework, take the midterms, and ace the final. The final grade will be determined by a formula.
Homework: Credit for homework is based on your submission of clear, concise, rigorous proofs. Homework is assigned every Thursday, and due the following Thursday, in class. Late homework will be acknowledged, but it may not be graded. You may work together on homework problems, but the set you submit must be original. Copying will anger the instructor.
Office Hours: (TBA)
Test Dates: If you can't make any of these dates, you must notify me weeks in advance.