Math 81/111: Abstract Algebra (Rings and Fields)
Winter 2015
Course Info:
 Lectures: Monday, Wednesday, Friday, block 10 (10:00 a.m.  11:05 a.m.)
 xperiod: Thursday 12:00  12:50 p.m.
 Dates: 5 January 2015  9 March 2015
 Room: 028 Haldeman
 Instructor: John Voight
 Office: Kemeny Hall, Room 341
 Email: jvoight@gmail.com
 Office hours: Monday 2:30  4:30 p.m., Tuesday 9:00  10:00 a.m., or by appointment
 Course Web Page: http://www.math.dartmouth.edu/~m111w15/
 Prerequisites: Math 71, or Math 31 and instructor permission
 Required Texts: J.S. Milne, Fields and Galois theory, version 4.50.
 Recommended Texts:
 David Dummit and Richard Foote, Abstract Algebra, Third edition, 2004. Roughly chapters 1314.
 Serge Lang, Algebra, Graduate Texts in Mathematics, vol. 211, Third edition, 2005. Roughly chapters IVVI.
 Ian Stewart, Galois Theory, Third edition, 2003.
 Grading: Grade will be based on weekly homework (50%) and a final exam (50%).
Syllabus:
[PDF] Syllabus
This course provides a foundation in core areas in the theory of rings and fields. Specifically, it provides an introduction to commutative ring theory with a particular emphasis on polynomial rings and their applications to unique factorization and to finite and algebraic extensions of fields. The study of fields continues with an introduction to Galois Theory, including the fundamental theorem of Galois Theory and numerous applications.
Homework:
The homework assignments will be assigned on a weekly basis and will be posted below. Homework is due in one week; no late homework will be accepted.
Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with youin other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment.
Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.
[PDF] Homework Submission Guidelines
1  5 Jan  (M)  Introduction  
2  7 Jan  (W)  1: Rings, fields  HW 1: PDF and TeX (updated 7 Jan); Solutions 
3  8 Jan  (R)  Review of Euclidean domains, PIDs, UFDs  
4  9 Jan  (F)  1: Polynomial rings, factoring, extension fields  
5  12 Jan  (M)  1: Algebraic and transcendental elements  
6  14 Jan  (W)  1: Constructions with straightedge and compass  HW 2: PDF and TeX; Solutions 
7  15 Jan  (R)  1: Algebraically closed fields  
8  16 Jan  (F)  2: Splitting fields  
  19 Jan  (M)  No class, Martin Luther King day  
9  21 Jan  (W)  2: Multiple roots  HW 3: PDF and TeX; Solutions 
10  22 Jan  (R)  Quadratic fields, examples  
11  23 Jan  (F)  2: Separability, 3: Automorphisms of fields  
12  26 Jan  (M)  3: Separable, normal, Galois extensions  
13  28 Jan  (W)  3: Fundamental theorem of Galois theory, examples  HW 4: PDF and TeX (updated 29 Jan); Solutions 
14  30 Jan  (F)  3: Fundamental theorem of Galois theory, proofs  
15  2 Feb  (M)  3: Fundamental theorem of Galois theory, consequences  
  4 Feb  (W)  No class, JV in Rochester  HW 5: PDF and TeX (updated 10 Feb); Solutions 
16  5 Feb  (R)  3: Examples  
  6 Feb  (F)  No class, Winter carnival  
17  9 Feb  (M)  3: Constructible numbers, Galois group of a polynomial; 4: A_n, transitive groups 

18  11 Feb  (W)  4: Cubic and quartic polynomials, Finite fields  HW 6: PDF and TeX; Sage code; Solutions 
19  12 Feb  (R)  4: Computing Galois groups over QQ  
20  13 Feb  (F)  5: Primitive element theorem  
21  16 Feb  (M)  5: Hilbert's Theorem 90  
22  18 Feb  (W)  5: Kummer theory  HW 7: PDF and TeX; Solutions (simplified solution to 7.3, 11 Mar) 
23  19 Feb  (R)  6: Algebraic closures  
24  20 Feb  (F)  5: Solvable extensions  
25  23 Feb  (M)  7: Topological groups, Krull topology  
26  25 Feb  (W)  7: Fundamental theorem of infinite Galois theory  HW 8: PDF and TeX; Solutions 
28  27 Feb  (F)  7: Fundamental theorem of infinite Galois theory, proofs  
  2 Mar  (M)  No class, JV in Bonn  
  4 Mar  (W)  No class, JV in Bonn  
29  6 Mar  (F)  Guest lecture by Peter Doyle  
30  9 Mar  (M)  Review and wrapup  
*  13 Mar  (F)  Final Exam, 8:00  11:00 a.m.; Solutions 