Instructor: Ljudmila Kamenova

e-mail: kamenova@math.sunysb.edu.
Office: Math Tower 3-115
Office hours: LK's web card
Grader: Alexandra Viktorova
Grader's office hours: Viktorova's web card

Feel free to send me an e-mail or drop by my office with questions.

The main goal of this course is to study in detail fundamental concepts and methods of algebra that are used in all branches of mathematics. During the second term we cover elements of homological algebra, field theory and foundations of algebraic geometry. We also study Galois theory and representations of finite groups.

Additional references:

• D. Cox, Galois Theory, Wiley-Interscience, 2004.
• M. Artin, Algebra, Prentice Hall, 1991.
• S. Lang, Algebra, 3^rd ed., Addison-Wesley, 1993.
• Jacobson, Basic Algebra, 2^nd ed, W.H. Freeman, New York, 1985, 1989.
• Hungerford, Algebra, Springer-Verlag, 1974.
• B. L. van der Waerden, Algebra, Springer-Verlag, 1994.
• Blyth, Module Theory, Oxford University Press, 1990.
• J.-P. Serre, Linear Representations of Finite Groups, Prentice Hall, 1991.

HW 1 (due on Feb 3): [DF] 17.1. Problems 2, 3, 4 and 5

HW 2 (due on Feb 10): [DF] 17.1. Problems 7, 10, 12 and 13

HW 3 (due on Feb 17): [DF] 13.1. Problem 8, 13.2. Problems 1, 7 and 10

HW 4 (due on Feb 24): [DF] 13.2. Problems 19, 20 and 21, 13.3. Problem 5

Spring Break: March 14--20, 2022

HW 5 (due on March 22): [DF] 13.4. Problem 5, 13.5. Problems 6 and 11, 13.6. Problem 8

Midterm: Tuesday, March 29, in class.

HW 6 (due on April 12): [DF] 14.2. Problems 3 (over Q), 7, 17 and 18

HW 7 (due on April 19): [DF] 14.3. Problems 3, 8 and 10 (p here is prime), 14.4. Problem 5

HW 8 (due on April 26): [DF] 14.6. Problems 5 (over Q), 11 and 19, 14.7. Problem 3

HW 9 (due on May 5): Click here for the problems.

Final: Tuesday, May 10, in Math Tower 4-130, 8:00am-10:45am.

1. Linear and multilinear algebra (4 weeks)
   • Minimal and characteristic polynomials. The Cayley-Hamilton Theorem.
   • Similarity, Jor`dan normal form and diagonalization.
   • Symmetric and antisymmetric bilinear forms, signature and diagonalization.
   • Tensor products (of modules over commutative rings). Symmetric and exterior algebra (free modules). Hom_R(- , -) and tensor products.

   References: Lang, chapters XIII and XIV; Dummit and Foote, Chapter 11.

2. Rudiments of homological algebra (2 weeks)
   • Categories and functors. Products and coproducts. Universal objects, Free objects. Examples and applications.
   • Exact sequences of modules. Injective and projective modules. Hom_R(- , -), for R a commutative ring. Extensions.

   References: Lang, chapter XX; Dummit and Foote, Part V, 17.

3. Representation Theory of Finite Groups (2 weeks)
   • Irreducible representations and Schur's Lemma.
   • Characters. Orthogonality. Character table. Complete reducibility for finite groups. Examples.

   References: Lang, chapter XVII; Dummit and Foote, Part VI; Serre.

4. Galois Theory (6 weeks)
   • Irreducible polynomials and simple extensions.
   • Existence and uniqueness of splitting fields. Application to construction of finite fields. The Frobenius morphism.
   • Extensions: finite, algebraic, normal, Galois, transcendental.
   • Galois polynomial and group. Fundamental theorem of Galois theory. Fundamental theorem of symmetric functions.
   • Solvability of polynomial equations. Cyclotomic extensions. Ruler and compass constructions

Student Accessibility Support Center Statement. If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Student Accessibility Support Center, ECC (Educational Communications Center) Building, Room 128, (631)632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential. https://www.stonybrook.edu/commcms/studentaffairs/sasc/facstaff/syllabus.php Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Student Accessibility Support Center. For procedures and information go to the following website: https://ehs.stonybrook.edu/programs/fire-safety/emergency-evacuation/evacuation-guide-people-physical-disabilities

Academic Integrity Statement. Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology & Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty please refer to the academic judiciary website at: http://www.stonybrook.edu/commcms/academic_integrity/index.html

Critical Incident Management Statement. Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of University Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Until/unless the latest COVID guidance is explicitly amended by SBU, during Fall 2021 "disruptive behavior" will include refusal to wear a mask during classes.