Math 603 COURSE INFORMATION - Spring 2017

TEXTS:

1. (Basic Text 1) Lang, S.: "Algebra," third edition, Springer-Verlag, U.S.A. (2002), ISBN 038795385X.
   
2. (Basic Text 2) Dummit, D.S. and Foote, M.: "Abstract algebra," third edition, Wiley, U.S.A., (2004), ISBN 0-471-43334-9.
   
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.

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, Todd Coxeter algorithm
9. Geometric applications, isometry groups, fundamental groups, pre sheaves, sheaves, derived functors, sheaf cohomology, Betti cohomology

B. Ring Theory

1. Definitions and basic examples
2. Isomorphisms, quotients, ideals.
3. Localization, Chinese remainder theorem.
4. Euclidean rings, Principal idŽal romains, Unique factorization domains
5. Discrete valuation rings, Dedekind rings.
6. The spectrum of a commutative ring, dimension theory
7. Polynomial rings, power series.

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 modules over a P.I.D., elementary divisors, Jordan and rational canoncial forms.
8. Homology.
9. Local rings, Nakayame Lemma, graded modules, Hilbert polynomial.

D. 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

E. Field Theory and Galois 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. Hilbert 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

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

GRADING:

               Homework                                 70% of grade 
               Final Exam                               30% of grade