Text: Algebra by Dummit & Foote
Class: MSC E408, TTh 10-11:15
Introduction: Abstract Algebra is the branch of pure mathematics that tries to reduce mathematical questions 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. That is an algebraic formulation of a geometric concept. ``Preserving the object's defining properties'' in this case means preserving the cube's shape (metric) 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. 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 modules. These abstract objects appear in practically every area of pure mathematics. We'll start with groups, then start on rings. We'll do modules and fields in Math 522.
Prerequisites: Undergraduate Abstract Algebra
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