• [Girard-Newton formulas] sums-of-powers in terms of elementary symmetric functions ... (updated Tue, 19 Jan 10, 05:19 PM)
• [Half-exactness of adjoint functors, Yoneda lemma] ... (updated Wed, 10 Dec 08, 02:58 PM)
• [Cantor-Schroeder-Bernstein theorem] ... (updated Sat, 19 Feb 05, 01:50 PM)
• [Simple algebras and involutions ] ... (updated Sat, 19 Feb 05, 01:50 PM)
• [Poincare-Birkhoff-Witt theorem ] ... (updated Fri, 08 Apr 11, 02:05 PM)
• [Cech vs. derived-functor sheaf cohomology] ... (updated Sat, 19 Feb 05, 02:00 PM)
• [Homology and derived functors ] ... (updated Tue, 14 Jun 11, 06:17 PM)

Course notes ... individual chapters from notes linked-to above:

• 01 the integers: unique factorization, integers mod m, Fermat's little theorem, Sun-Ze's theorem, examples.
• 02 groups I: subgroups, Lagrange's theorem, homomorphisms, kernels, normal subgroups, cyclic groups, quotient groups, groups acting on sets, Sylow theorems, worked examples
• 03 the players: rings, fields, etc: homomorphisms, vectorspaces, modules, algebras, polynomial rings I
• 04 commutative rings I: divisibility and ideals, polynomials in one variable over a field, ideals and quotients, maximal ideals and fields, prime ideals and domains, Fermat-Euler on sums of two squares, examples
• 05 linear algebra I: dimension, bases, homomorphisms
• 06 fields I: adjoining things, fields of fractions, fields of rational functions, characteristics, finite fields, algebraic field extensions, algebraic closures
• 07 some irreducible polynomials: over a finite field, examples
• 08 cyclotomic polynomials: multiple factors in polynomials, finite subgroups of fields, infinitude of primes p=1 mod n, examples
• 09 finite fields: uniqueness, Frobenius automorphism, counting irreducibles
• 10 modules over PIDs: structure theorem, variations, finitely-generated abelian groups, Jordan canonical form, conjugacy versus k[x]-module isomorphism, examples
• 11 finitely-generated modules: free modules, finitely-generated modules over a domain, PIDs are UFDs, structure theorem (again), submodules of free modules
• 12 polynomials over UFDs: Gauss' lemma, fields of fractions, examples
• 13 symmetric groups: cycles, disjoint cycle decomposition, transpositions, examples
• 14 naive set theory: sets, posets, ordinals, transfinite induction, finiteness/infiniteness, comparison of infinities, transfinite Lagrange replacement, equivalents of the Axiom of Choice
• 15 symmetric polynomials: discriminants, examples
• 16 Eisenstein's criterion: examples
• 17 Vandermonde determinants: examples
• 18 cyclotomic polynomials II: over the integers, examples
• 19 roots of unity: cyclotomic fields, solutions in radicals, Lagrange resolvents, quadratic fields, quadratic reciprocity, examples
• 20 cyclotomy III: prime power cyclotomic polynomials over the rationals, irreducibility, factoring, examples
• 21 primes in arithmetic progressions: Euler and the zeta function, Dirichlet's theorem, dual groups of abelian groups, non-vanishing on the line Re(s)=1, analytic continuations, Dirichlet series with positive coefficients
• 22 Galois theory: field extensions, imbeddings, automorphisms, separable field extensions, primitive elemenets, normal extensions, Galois' theorem, conjugates, trace, norm, examples
• 23 solution by radicals: Galois' criterion, composition series, Jordan-Holder theorem, solving cubics by radicals, examples
• 24 eigenvectors, eigenvalues, spectral theorems: diagonalizability, semi-simplicity, commuting endomorphisms ST=TS, inner product spaces, projections without coordinates, unitary operators, spectral theorems, corollaries, examples
• 25 duality, naturality, bilinear forms: dual vectorspaces, examples of naturality, bilinear forms, examples
• 26 determinants I: prehistory, definitions, uniqueness, existence
• 27 tensor products: desiderata, uniqueness, existence, tensor products of maps, extension of scalars, functoriality, naturality, examples
• 28 exterior algebra: desiderata, uniqueness, existence, exterior powers of free modules, determinants revisited, minors of matrices, uniqueness in the structure theorem, Cartan's lemma, Cayley-Hamilton theorem, examples