Math 1530: Abstract algebra, Spring 2017

Class  Date  Topics  Reading 
1  Thurs Jan 26  Sets, functions, binary operations. Definition of a group  p.1, p.1516, this article 
2  Tues Jan 31  Equivalence relations. The group Z/nZ  p. 3, p. 89 
3  Thurs Feb 2  (Z/nZ)* and arithmetic in Z/nZ. Some properties of groups. Order  p. 10, p. 1820 
4  Tues Feb 7  Injective, surjective, bijective functions. Dihedral groups, symmetric groups.  p. 2, pp. 2325 
5  Thurs Feb 9  Symmetric groups. Homomorphisms, isomorphisms. (Video lecture due to snow)  pp. 2932, pp. 3639 
6  Tues Feb 14  Homomorphisms, subgroups, generators.  parts of p. 36, 4648, 50, 6163 
7  Thurs Feb 16  Greatest common divisor and least common multiple; Euclidean algorithm. Cyclic groups  pp. 45, 5456 
8  Thurs Feb 23  Group actions. Examples, permutation representation. (Guest lecture, Michael Zellinger)  pp. 4144, 112114 
9  Tues Feb 28  Cyclic groups again, lattice of subgroups. Image and kernel. Cosets.  pp. 5759, 6667, 
10  Thurs March 2  Cosets, normal subgroups, conjugation and conjugacy classes.  pp. 77, 82, 89, 123 
11  Tues March 7  Normal subgroups again; kernels again. Definition of the quotient group. First isomorphism theorem  pp. 8183, 97 
12  Thurs March 9  The isomorphism theorems.  pp. 98100 
13  Tues March 14  Snow day  
14  Thurs March 16  Simple groups. Composition series and the Jordan Holder theorem. Signs of permutations; the alternating group.  pp. 97, 101103, 106110 
15  Tues March 21  Midterm pep talk. Direct products; the fundamental theorem of finitely generated abelian groups. Semidirect products.  pp. 152155, 158159, 171, 175176. 
16  Thurs March 23  Semidirect products, Sylow theorems (statement).  
17  Tues April 4  Rings, basic properties. Examples: polynomial rings, matrix rings, group rings. Definition of integral domain and field.  7.1, 7.2 
18  Thurs April 6  Homomorphisms, ideals, and quotients of rings.  7.3 
19  Tues April 11  Ideals generated by elements. Principal ideals; ideals of Z. Maximal ideals. (Guest lecture Michael Zellinger)  7.4 
20  Thurs April 13  Maximal ideals and prime ideals. Sums, products, intersections of ideals.  end of 7.3, 7.4 
21  Tues April 18  Norms, Euclidean domains, division algorithm. Euclidean domains are principal ideal domains.  8.1 
22  Thurs April 20  Principal ideal domains, unique factorization domains. The field of fractions of an integral domain.  8.2, 8.3 
23  Tues April 25  PIDs are UFDs. Polynomial rings over UFDs are UFDs.  8.3, 9.3 
24  Thurs April 27  Fields. Characteristic and prime subfield. Field extensions and degree. Adjoining a root of a polynomial.  13.1 
25  Tues May 2  Algebraic field extensions.  13.1, 13.2 
26  Thurs May 4  Algebraic field extensions. Impossibility of trisecting angles etc.  13.2, 13.3 
Collaboration
You are welcome to collaborate with other students in the class on your homework, although I suggest that you think carefully about each problem on your own first. You are required to write up your solutions separately and write the names of the students with whom you worked on the assignment. (You may only use the Internet as a general reference, at the level of generality of Wikipedia.)
How much time will this class take?
Roughly speaking, you should spend ten hours every week outside of class, including attending Michael's problem session, my office hours, reviewing class material and doing homework. Attending the TA problem session is strongly encouraged. In addition to three hours of class every week and about twenty hours of additional exam preparation, I estimate a total of 189 hours of time spent on this class (39 class hours, 130 studying hours and 20 exam studying hours).
Accommodations for students with disabilities
Any student with a documented disability is welcome to contact me as early in the semester as possible so that we may arrange reasonable accommodations. As part of this process, please be in touch with Student and Employee Accessibility Services by calling 4018639588 or online at
http://brown.edu/Student_Services/Office_of_Student_Life/seas/index.html