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.)
- x-period: 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
- E-mail: 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 13-14.
- Serge Lang, Algebra, Graduate Texts in Mathematics, vol. 211, Third edition, 2005. Roughly chapters IV-VI.
- 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 you--in 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 straight-edge 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 |
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| 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 wrap-up | |
| * | 13 Mar | (F) | Final Exam, 8:00 - 11:00 a.m.; Solutions |