Topics Covered:
Groups, cont.:WRITING UP HOMEWORK SOLUTIONS: You are expected to show all essential steps in solving a problem. Draw appropriate figures. If in doubt, PUT IT IN! Make your solution flow. Ask yourself the question: "If I were the reader, how would I score this problem?'' Numerical answers alone, single words "yes" or "no", etc., are NEVER, EVER acceptable as complete answers. One of the skills you are expected to learn in this class is that of communicating mathematics clearly and completely in writing. Writing up homework exercises is your opportunity to hone these skills. Your grade depends on it!
Series of groups, composition series, Jordan-Hölder Theorem, Sylow theorems,
class equation, finite simple groups, classification problems, generalized
Cayley theorem, index theorem, the simplicity of A_5, groups of small order.
Rings & Fields:
Definitions, examples, subrings, integral domains, fields, field of quotients,
polynomial rings, polynomial factorization, non-commutative rings. Ring
homomorphisms, factor rings, prime & maximal ideals, extension fields, vector
spaces, algebraic extensions, geometric constructions, finite fields.
Homework Sets:
1. p. 310: 4,6,7,9
2. p. 326: 6,11,13,18,22
3. p. 332: 2,4,8
4. p. 319: 4,9,20,22
5. p. 175: 12,18,38,44,50
6. p. 182: 4,14,23,28
7. p. 197: 4,12,14
8. p. 207: 4,22,24
9. p. 218: 4,12,28,35