Algebra Syllabus

Category theory I:

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.

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.

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.

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.

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).

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.

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.

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).

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, \(S_3\), \((S_4)\).

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