Tue & Thu 9:45am-11:05am in Math Tower 4-130

**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.

- 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.

- 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.

- 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.

- 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
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**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.