Abstract algebra (a graduate course)

Instructor: M. Grützmann (古梅西, e-mail: x@nwpu.edu.cn, x=grutzmann, QQ: 109-235-1405)
Textbook: P.A. Grillet, Abstract Algebra, GTM, vol.242, Springer (2007), ISBN 978-0-387-71568-1. A summary can be found here (it will be extended as the semester proceeds).
Instruction language: English
Meeting time and place: Mon 1900–2040 at 教东C201 (Chang'an campus), We 1600–1740 at 教东C203;
Final: 2013, Jan 14, 2:30-4:30pm (120min), 教东楼 B108 or Jan 15, 8:30–11:59am (120min), TBA
Course home page: www.grutzmann.de/algebra.


Please send your name and the word algebra when you add me.

Description

This is a graduate course in abstract algebra. It covers finite group theory including Sylow's theorems, normal series, and representations; (commutative unital) ring theory including domains and fields, principal ideal domains, unique factorization domains; and Galois theory including algebraic and transcendental extensions, algebraic closure, separable and normal extensions, Galois' correspondence principle, solvability (by radicals), geometric constructability, and some more.

Grading

The course grade constitutes from homework (10%), a 50min talk (30%), and the final exam (60%).

Homework will be assigned every week and is due on Monday the next week. The graded homework will be returned within 1.5 weeks.

Presentations by the students

A list of topics for talks together with points that should be included is given here. Generally the talks should not exceed 1h. Either you prepare some handout for your classmates or you write important things on the black/ white board. If you give me the talk 2 days before you hold it, I can double check it and give suggestions for improvement if needed.

Final Exam

The final exam will be in the last week of the semester Jan 14–18 (2013). It is designed for 120min, based on the material covered in class, i.e. Finite groups (definitions–semi-direct products), (commutative unital) Rings (definitions–unique factorization domains/ localizations), and Field extensions/ Galois theory (definition–geometric constructions). You cannot use text-books or computers/ calculators in the final, but a dictionary/ vocable list as well as a 5 pages “summary” is permitted in the exam. Place and starting time will be announced here (and in class) when fixed. There will also be some typical questions linked here. In the mean-time you can search the web for algebra questions in the math part of the GRE test/ comprehensive exam algebra (another one).

Homework

0 please send me an e-mail with your name (汉字+PY) and which topic you would like to present by Oct 3rd. solutions

1 Groups

1st Homework due on Oct 15th. solutions

2nd Homework due on Oct 22nd. solutions

3rd Homework Section 1.5 due on Oct 29th. (The rest the week after.) Nr. 1.5.1b had an error, i.e. the given definition is the free coproduct. In addition you can use ρG(gH(h) = ρH(hG(g) for all g∈ G, h∈ H. solutions

4th Homework due on Nov 5th. solutions

5th Homework due on Nov 12th. solutions

6th Homework due on Nov 19th. solutions

7th Homework due on Nov 26th. solutions

2 Rings and algebras

8th Homework due on Dec 3rd. Erratum: Please note that an earlier version of this assignment sheet contained a wrong definition of the product in the unitalization R1:=ZR. The correct version is (m,a)(n,b) = (mn, ab+mb+na) for m,nZ and a,b∈ R (where na = a+a+…+a, n times for n ≥ 0 and (-n)a=-(na) for the other cases).

9th Homework due on Dec 10th.

10th Homework due on Dec 17th. Erratum: an earlier version of Exercise 2.5.3 asked you to show the localization trick for prime ideals pR. Actually if you have a short proof, then it probably also extends to arbitrary ideals I⊲R. In Exercise 2.5.4 you can use the result for arbitrary ideals.

3 Field extensions and Galois theory

11th Homework due on Dec 24th.

12th Homework due on Jan 2nd.

Last Homework due on Jan 9th.

Please try to finish as much homework as possible by Jan 11th, especially you should do at least one homework sheet from chapter 3. I will ignore those homework wich you could not do in time, but the material could still be part of the exam. I will try to put online the remaining solutions by evening Jan 11th.


Table of contents (tentative)

The numbers in parenthesis give the approximate number of lectures.

  1. Groups (5+ weeks)
    1. Definition, Examples, Subgroups, and Homomorphisms (5)
    2. Isomorphism theorems (2)
    3. Free groups, free products, and presentations (2)
    4. Direct products & The Krull–Schmidt theorem (2)
    5. Group actions (1)
    6. Structure of symmetric groups (1)
    7. The Sylow theorems (1)
    8. Small groups (1)
    9. The general linear group (1)
    10. Group representations and characters (1.5)
    11. Composition series and Jordan–Hölder theorem (1)
    12. Group extensions and Solvable groups (1*)
    13. Semidirect products and Nilpotent groups (1*)

  2. Rings and algebras (3 weeks)
    1. Definition, Examples, Homomorphisms, Subrings, and Ideals
      integers, integers modulo n, polynomials, matrix algebras, direct sum/product, tensor product, … (2)
    2. Domains and fields (1)
    3. Principal ideal domains (1)
    4. Unique factorization domains (1)
    5. Noetherian rings (1*)
    6. Gröbner bases (1*)
    7. Localizations (1)
    8. Dedekind domains (1*)
    9. Minimal prime ideals and Krull dimension (1*)

  3. Field extensions and Galois theory (5 weeks)
    1. Algebraic and transcendental extensions (1)
    2. The algebraic closure (1)
    3. Separable extensions (1)
    4. Resultants and discriminants (1)
    5. Splitting fields and Normal extensions (1.5)
    6. Galois extensions and the correspondence principle (1)
    7. Infinite Galois extensions and Picard–Vessiot theory (2*)
    8. Cyclotomic and cyclic extensions and Solvability by radicals (1)
    9. Norm and trace (1)
    10. Geometric constructions (1)
    11. Algebraic integers (1*)
    12. Outlook: Algebraic geometry (1–3)
    13. Ordered fields and valuations (1*)
    14. Extensions II and valuations? (1*)
    15. Hensel's Lemma (1*)
    16. Filtrations and completions? (1*)

  4. Outlook: Abstract nonsense (2 weeks*)
    1. Categories and additive categories: definition, examples, functors (2)
    2. Limits and colimits (1)
    3. Tensor products: tensor algebra, symmetric algebra, exterior algebra (1.5)
    4. Completions (1*)
    5. Homomorphisms (1)
    6. Adjoint functors (with theorem; 1.5)
    7. Triples (1)

* as time permits.

Further literature

  1. M. Artin: Algebra, 2nd edition, Addison Wesley (2010), ISBN 978-013-241-377-0.
  2. N. Jacobson: Lectures in abstract algebra: 1 basic concepts, 3 Fields and Galois theory, Springer, (2000), ISBN 750-620-060-0, 750-620-062-7.
  3. 里克正 (Li Kezheng): 抽象代数基础 (Basic Algebra), Springer (2007)/ Higher Education Press, ISBN 978-730-214-407-6.
  4. J.J. Rothman: A first course in abstract algebra, 3rd edition, Prentice Hall (2006), ISBN 978-711-121-262-1.
  5. 杨子胥 (Yang Zixu): 近世代数 (International Algebra), 3rd edition, Higher Education Press (2011), ISBN 978-704-030-072-7.

This page is maintained by M. Grützmann.