Math 502-Math 503 COURSE INFORMATION - Fall 2019

TEXTS:

1. (Basic Text 1) Dummit, D.S. and Foote, M.: "Abstract algebra," third edition, Wiley, U.S.A., (1999), ISBN ISBN 0-471-43334-9.
   
   
2. (Basic Text 2) 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. (Optional) 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 (for two semesters, i.e. for both math 502 and math 503)

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

I will post reading and homework assignments on the web, and there will be a final exam in this couse.

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                                 40% of grade
               Mid-term                                 15% of grade  (Oct. 3)
               Mid-term                                 15% of grade (Nov. 7)
               Final Exam                               30% of grade (Dec. 18, assuming one has a passing score)