This page will be updated at least once a week. Please check it regularly for announcements.

Announcement: Solutions to problems 4,5 of the final are posted.

Classes: TuTh 12:50pm- 2:10pm Earth&Space 181 Instructor: Valentina Kirichenko (office Math 3-102, phone 632-8884, e-mail: vkiritch at math dot sunysb dot edu) Webpage: http://www.math.sunysb.edu/~vkiritch/ Office Hours: TuTh 3-4pm and 5-6pm

Grader: Jaimal Thind (e-mail: jthind at math dot sunysb dot edu) Webpage: http://www.math.sunysb.edu/~jthind/ Office Hours: TBA

About this Course: 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 linear and multilinear algebra, field theory and foundations of algebraic geometry. We also study Galois theory and representations of finite groups.

Additional reading:

• M. Artin, Algebra, Prentice Hall, 1991. (There are two copies in the Math/Physics library on reserve.)

S. Lang, Algebra, 3rd ed., Addison-Wesley, 1993. Jacobson, Basic Algebra, 2nd 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, Springer-Verlag, 1977

Homework/Class Participation 30%, Midterm Tests 15%+15%, Final Exam 40%

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

January 31-February 2
Section 12.1 Artin: Chapter 12; Proof of the decomposition theorem.
Homework 1 due February 9

February 7-9
Sections 12.2, 12.3; Rational and Jordan canonical forms; Minimal and characteristic polynomials; Cayley-Hamilton theorem.
Homework 2 due February 16

February 14-16
Sections 11.3-11.5; Dual vector spaces. Lagrange interpolation formula. Tensor, symmetric and exterior algebras. Determinants.
Homework 3 due February 23

February 21-23
Artin: Chapter 7; More on symmetric forms. Polarizations of polynomial functions. Inner products. Gram-Schmidt orthogonalization. Geometry in Euclidean spaces. Spectral decomposition for normal operators.
Homework 4 due March 2

February 28-March 2
Sections 17.1 and 10.5; Introduction to homological algebra: the long exact sequence in cohomology. Exact sequences of modules. Injective and projective modules. Functors Hom(D , -) and Hom(- , D).

March 7-9
No classes. Solve the take home midterm test.
Midterm test due March 16

March 14-16
Sections 17.1, 15.2, 15.3 and 13.1, 13.2; Extensions (cohomology derived from Hom(- , D)). A little bit of algebraic geometry: Hilbert's Nullstellensatz. Galois theory: basic definitions. Examples of fields, field extensions and degree of a field extension.
Homework 5 due March 24

March 21-23
Section 13.3,13.6; Classical straightedge and compass constructions. Splitting fields: cyclotomic extensions. Construction of regular n-gons. Trisection of an angle.
Homework 6 due March 31

March 28-30
Section 13.4,13.5; Automorphisms of a field extension. Galois theory for cyclotomic extensions. Separable and normal extensions. Existence and uniqueness of splitting fields. Finite fields.
Homework 7 due April 7

April 4-6
Section 14.1,14.2,14.5; The Primitive Element Theorem. Galois extensions. The Fundamental Theorem of Galois Theory. Applications to cyclotomic extensions.
Homework 8 due April 21

April 18-20
Section 14.3,14.6,14.7; Examples: Galois group of a cubic, Abelian extensions. Solvability by radicals.
Homework 9 due April 28

April 25-27
Midterm test and discussion of solutions.

May 2-4
Sections 18.1-18.3, see also Artin; Representation theory. Mashke's theorem, Schur's lemma, orthogonality of characters.
Optional practice problems about group representations