Math 371 COURSE INFORMATION - Fall 2014

Faculty: Ted Chinburg
Office Hours: In the evening via Skype at times to be arranged in class
TEXTS:
  1. Dummit, D.S. and Foote, M.: "Abstract algebra," third edition, Wiley, U.S.A., (1999), ISBN ISBN 0-471-43334-9.

  2. (Optional text 1) Lang, S.: "Algebra," 3rd rev. ed. 2002. Corr. 4th printing, 2004, XV, Graduate Texts in Mathematics, Vol. 211, Springer, ISBN: 0-387-95385-X

  3. (Optiona text 2) Artin, M.: "Algebra," Prentice Hall, U.S.A., (1991) ISBN 0-13-004763-5.


Comments: The Dummit and Foote text provides a more compact but less complete presentation of the text than does Lang's book. Artin's book has a better discussion of matrix groups and of some other topics, such as groups defined by generators and relations. I will suggest readings from all three texts, as well as from some journal articles, for different parts of the course.

SYLLABUS

A. Group Theory
  1. Definitions and basic examples.
  2. Subgroups, normality, centralizers, normalizers, quotients.
  3. Isomorphism Theorems, composition series, Jordan-Holder, simple groups.
  4. Sylow Theorems, class equation, applications.
  5. Solvable and nilpotent groups
  6. Finite products, semi-direct products, group extensions
  7. Categories, products, coproducts.
  8. Products and coproducts of groups, free groups, generators and relations.
  9. Geometric applications, isometry groups, fundamental groups.

B. Ring Theory
  1. Definitions and basic examples
  2. Homemorphisms, quotients, ideals.
  3. Localization, Chinese remainder theorem.
  4. The spectrum of a commutative ring.
  5. Polynomial rings, power series.
  6. Discrete valuation rings, Dedekind rings.

C. Module Theory.
  1. Definitions and basic examples.
  2. Homomorphisms, quotients, direct sums and products.
  3. Noetherian rings and modules, Hilbert basis theorem, power series, Artinian modules.
  4. Vector spaces, dual spaces, determinants.
  5. Projective, injective and flat modules.
  6. Direct and inverse limits.
  7. Finitely generated odules over a P.I.D., elementary divisors, Jordan and rational canoncial forms.

D. Field Theory
  1. Definitions, examples, finite and algebraic extensions.
  2. Algebraic closure, splitting fields, normal extensions.
  3. Separable and inseparable extensions.
  4. Galois extenions, Galois groups, fundamental theorem of Galois theory.
  5. Examples (finite fields, cyclotomic fields, Kummer extensions, computation of Galois groups).
  6. Norms and traces.
  7. Cyclic, solvable, radical, abelian extensions. Applications.
  8. Hilbet Theorem 90, Normal basis theorem, algebraic independence of characters.
  9. Infinite extensions, Z_p extensions, transcendental extensions, Kummer theory.
  10. Integral extensions of rings, completions and absolute values.

COURSE ORGANIZATION

GRADING:

               To pass the course, it is necessary to pass the final exam.
               Assuming one passes this exam, the course grade will be computed
               in the following way:
               
               Homework                                 55% of grade
               Mid-term                             15% of grade 
               Final Exam                                30% of grade (assuming one has a passing score)
               
Last updated: 8/24/14
Send e-mail comments to: ted@math.upenn.edu