Math 602 COURSE INFORMATION - Fall 2013

Faculty: Ted Chinburg
Office Hours: By appointment
TEXTS:
  1. (Basic Text 1) Lang, S.: "Algebra," third edition, Springer-Verlag, U.S.A. (2002), ISBN 038795385X.

    Remarkably, this book can now be downloaded for free as a .djvu file: http://bookos.org/g/Serge%20Lang . I would be careful about checking any files of this kind for viruses, though. You can also buy a hard copy for $67.82 from Amazon. The book may be available at the Penn bookstore, but keep in mind that the bookstore marks up all books by 33 per cent.

  2. (Basic Text 2) Dummit, D.S. and Foote, M.: "Abstract algebra," third edition, Wiley, U.S.A., (2004), ISBN 0-471-43334-9.

    A google search will show various web sites which claim to have free downloads of this book, but be careful about viruses. Amazon advertises the book for $70.83, which is far lower than the price of Wiley.

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

    This book is available free at http://bookos.org/g/algebra+textbooks . One can find $15 hard cover versions, e.g. at http://www.abebooks.com/Algebra-Michael-Artin-Prentice-Hall/9009201250/bd


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.

COMBINED SYLLABUS FOR MATH 602 AND MATH 603.

Note: Math 602 covers parts A, B and half of C.

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.
  8. Homology.
  9. Local rings, Nakayame Lemma, graded modules, Hilbert polynomial.

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.

E. Linear Algebra
  1. Simplicity and semi-simplicity.
  2. Density Theorem, Wedderburn Theorem
  3. Matrices and bilnear forms. Symmetric and hermitian forms.
  4. Structure of bilinear forms
  5. Spectral Theorem
  6. Tensor product, symmetric product, alternating product.
  7. Koszul complex, Hilbert Syzygy Theorem, derivations.
  8. Representations of groups, characters

COURSE ORGANIZATION

GRADING:
               Homework                                 70% of grade 
               Final Exam                               30% of grade 
Last updated: 8/27/13
Send e-mail comments to: ted@math.upenn.edu