Algebra Syllabus

SYLLABUS FOR ALGEBRA, MATH 470

1st QUARTER

Category theory I:

Definitions, initial/final objects, (co)products.

Groups:

Notions: Definitions, normal subgroups, actions, nilpotent, solvable, simple, extensions, semi-direct product, composition series, presentation, Cayley graphs.

Theorems: Isomorphism theorem, Lagrange, Jordan-Hölder, Sylow.

Examples: \(S_n\), \(A_n\), \(GL(n,p)\), \(GL(n R)\), \(U_n\), free groups, \(p\)-groups.

Rings and Modules:

Notions: Definitions, ideals, integral domains, prime and maximal ideals, Noetherian rings, complexes, homology, exact sequences.

Theorems: Isomorphism theorem, snake lemma, Hilbert's basis theorem.

Examples: Polynomial rings, power series, \(p\)-adics.

2nd QUARTER

Factorization in rings:

Notions: prime and irreducible elements, PID's, UFD's, Euclidean domains.

Theorems: Polynomial rings over UFD's are UFD's, Chinese Remainder Theorem.

Examples: irreducibilty in polynomial rings, Eisenstein's criterion, Gaussian integers.

Linear algebra I:

Notions: bases and dimension, torsion, presentations and resolutions, characteristic polynomials, canonical forms.

Theorems: classification of finitely generated modules over PID's, Cayley-Hamilton, Jordan canonical form.

Fields:

Notions: definition, simple extensions, finite and algebraic extensions, algebraic closure, splitting field, normal and separable extensions, Galois extensions, (transcendental extensions in 3rd quarter).

Theorems: existence of algebraic closure, Galois correspondence.

Examples: rational functions, finite fields, cyclotomic fields, applications of Galois theory (e.g. solvability of polynomials).

3rd QUARTER

Category theory II:

Notions: Functors, natural transformation, exact functors, adjoint functors, limits, colimits (and filtered colimits), relation with universal properties.

Theorems: Preservation of limits/colimits by adjoint functors.

Linear algebra II:

Notions: Tensor product, extension of scalars, symmetric and exterior powers, flat modules, projective modules, injective modules, Tor, Ext.

Theorems: A free module is projective. A projective module is flat. Existence and homotopy uniqueness of projective and injective resolutions of modules.

Commutative algebra:

Notions: Localization of rings and modules, integral homomorphism of rings, integral closure, normal domains, spectra of rings and Zariski topology, transcendental field exensions, Krull dimension.

Theorems: Nakayama's Lemma, Going-up and Going-down, Noether normalization, Hilbert and Zariski Nullstellensatz, (transcendence degree equals Krull dimension for finitely generated algebras over a field).

Homological algebra:

Notions: Additive category, abelian category, complexes, homotopy category of complexes, derived category, derived functors and cohomology, spectral sequence.

NOTE: At the discretion of the instructor, the Homological Algebra section, and possibly the final topics in the Commutative Algebra section, could be replaced by:

Representation theory of finite groups (char. 0):

Notions: Irreducible representation, group algebra, matrix coefficients, character, induction, Fourier transform, convolution.

Theorems: Maschke, Schur's lemma, Frobenius's reciprocity, orthogonality relations, Plancherel.

Examples: Abelian groups, \(S_3\), \((S_4)\).

Noncommutative algebra:

Structure of semisimple rings, Jacobson radical, Wedderburn Theorem, division algebras.