MAT 534 and 535 Algebra I + II

Prerequisites

A year of undergraduate algebra, such as MAT 313 and MAT 318. Thus basic notions concerning set theory, cardinals, ordinals, prime numbers, Euclidean algorithm, congruences, polynomials, complex numbers, abelian and cyclic groups, permutation groups, rings and fields, vector spaces are assumed or briefly reviewed. A good reference is Algebra by Michael Artin, Prentice Hall, 1991.

Algebra I (Fall)

1. Groups (5 weeks)
   • Direct products, Normal subgroups, Quotient groups, and the isomorphism theorems.
   • Groups acting on sets; orbits and stabilizers. Applications: class formula, centralizers and normalizers, centers of finite p-groups. Conjugacy classes of S_n.
   • Sylow's Theorems, Solvable groups, Simple groups, simplicity of A_n. Examples: Finite groups of small order (< 8).
   • Structure of finitely generated abelian groups. Free groups. Applications.
   
   References: Algebra (3rd Edition), Lang, 1993, Addison-Wesley, chapter I. Abstract Algebra (2nd edition), Dummit and Foote, 1999, Part I. Introduction to the Theory of Groups, Rotman, Springer Verlag.

2. Basic linear algebra (3 weeks)
   • Vector spaces, Linear dependence/independence, Bases, Matrices and linear maps. Dual vector space, quotient vector spaces, isomorphism theorems.
   • Determinants, basic properties. Eigenspaces and eigenvectors, characteristic polynomial.
   • Inner products and orthonormal sets. Spectral theorem for normal operators (finite dimensional case).
   
   References: Algebra (3rd Edition), Lang, 1993, chapters XIII and XIV. Abstract Algebra (2nd Edition), Dummit and Foote, Chapter 11.

3. Rings, modules and algebras (6 weeks)
   • Rings, subrings, fields, ideals, homomorphisms, isomorphism theorems, polynomial rings.
   • Integral domains, Euclidean domains, PID's. UFD's and Gauss's Lemma (F[x₁, . . . , x_n] is an UFD). Examples.
   • Prime ideals, maximal ideals. The Chinese remainder Theorem. Fields of fractions.
   • The Wedderburn Theorem (no proof). Simplicity and Semisimplicity.
   • Noetherian rings and the Hilbert Basis Theorem.
   • Finitely generated modules over PID's, the structure theorem.
   
   References: Algebra (3rd Edition), Lang, 1993, Addison-Wesley, chapters II, III, V and VI. Basic Algebra (2nd edition) Jacobson, Chapter 2. Abstract Algebra (2nd Edition), Dummit and Foote, Part II.

Algebra II (Spring)

1. Linear and multilinear algebra (4 weeks)
   • Minimal and characteristic polynomials. The Cayley-Hamilton Theorem.
   • Similarity, Jordan 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: Algebra (3rd Edition), Lang, 1993, chapters XIII and XIV. Abstract Algebra (2nd Edition), 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: Algebra (3rd Edition), Lang, 1993, chapter XX, Dummit and Foote, 1999, 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: Algebra (3rd Edition), Lang, 1993, Addison-Wesley, chapter XVIII. Linear representations of finite groups, J.-P. Serre, 1977, Springer-Verlag. Abstract Algebra (2nd edition), Dummit and Foote, Part VI.

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.
   
   References: Algebra (3rd Edition), Lang, 1993, chapters VII and VIII. Galois Theory, Emil Artin. Abstract Algebra (2nd edition), Dummit and Foote, 1999, Part IV.

General References

• Algebra (3rd edition), S. Lang, 1993, Addison-Wesley
• Abstract Algebra (2nd edition), Dummit and Foote, 1999, John Wiley.
• Algebra, Hungerford, 1974, Springer-Verlag
• Basic Algebra (2nd edition) N. Jacobson, W.H. Freeman, New York, 1985, 1989.
• Algebra, B.L. van der Waerden, Springer-Verlag, 1994.
• Module Theory, Blyth, 1990, Oxford University Press