[Course Notes] ... (updated Sat, 21 Jul '07, 12:39 PM) ... [Table of Contents] ... in individual chapters below.

[linked PDF version!] (thanks to Iver Walkoe!) ... [ Index]

Other miscellaneous notes: Solutions to standard exercises: s01 , s02 , s03 , s04 , s05 , s06 , s07 , s08 , s09 , s10 , s11 , s12 , s13 , s14 , s15 , s15b , s16 , s17 , s18 , s19 , s20 , s21

** 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

Elementary exercises and notes: [Intro to Abstract Algebra]

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