Abstract Algebra (Hardcover)

PREFACEINTRODUCTION THE INTEGERSUnique factorization IrrationalitiesZ/m, the integers mod m Fermat's little theorem Sun-Ze's theorem Worked examplesGROUPS IGroups Subgroups Homomorphisms, kernels, normal subgroupsCyclic groups Quotient groups Groups acting on sets The Sylow theorem Trying to classify finite groups, part I Worked examples THE PLAYERS: RINGS, FIELDSRings, fields Ring homomorphisms Vector spaces, modules, algebras Polynomial rings I COMMUTATIVE RINGS IDivisibility and idealsPolynomials in one variable over a fieldIdeals and quotients Ideals and quotient rings Maximal ideals and fields Prime ideals and integral domains Fermat-Euler on sums of two squares Worked examplesLINEAR ALGEBRA I: DIMENSIONSome simple results Bases and dimension Homomorphisms and dimension FIELDS I Adjoining things Fields of fractions, fields of rational functions Characteristics, finite fields Algebraic field extensions Algebraic closures SOME IRREDUCIBLE POLYNOMIALS Irreducibles over a finite field Worked examplesCYCLOTOMIC POLYNOMIALSMultiple factors in polynomials Cyclotomic polynomials Examples Finite subgroups of fields Infinitude of primes p = 1 mod n Worked examples FINITE FIELDS Uniqueness Frobenius automorphismsCounting irreducibles MODULES OVER PIDSThe structure theorem Variations Finitely generated abelian groups Jordan canonical form Conjugacy versus k[x]-module isomorphismWorked examples FINITELY GENERATED MODULES Free modulesFinitely generated modules over a domain PIDs are UFDs Structure theorem, again Recovering the earlier structure theorem Submodules of free modules POLYNOMIALS OVER UFDS Gauss's lemma Fields of fractions Worked examples SYMMETRIC GROUPS Cycles, disjoint cycle decompositions Transpositions Worked examples NAIVE SET THEORYSets Posets, ordinals Transfinite induction Finiteness, infiniteness Comparison of infinitiesExample: transfinite Lagrange replacement Equivalents of the axiom of choice SYMMETRIC POLYNOMIALS The theorem First examples A variant: discriminantsEISENSTEIN'S CRITERION Eisenstein's irreducibility criterion Examples VANDERMONDE DETERMINANTS Vandermonde determinants Worked examples CYCLOTOMIC POLYNOMIALS II Cyclotomic polynomials over ZWorked examples ROOTS OF UNITY Another proof of cyclicness Roots of unity Q with roots of unity adjoinedSolution in radicals, Lagrange resolvents Quadratic fields, quadratic reciprocity Worked examplesCYCLOTOMIC III Prime-power cyclotomic polynomials over QIrreducibility of cyclotomic polynomials over QFactoring Fn(x) in Fp[x] with p|n Worked examples PRIMES IN ARITHMETIC PROGRESSIONS Euler's theorem and the zeta function Dirichlet's theoremDual groups of abelian groups Non-vanishing on Re(s) = 1 Analytic continuations Dirichlet series with positive coefficients GALOIS THEORY Field extensions, imbeddings, automorphisms Separable field extensionsPrimitive elements Normal field extensions The main theorem Conjugates, trace, norm Basic examplesWorked examples SOLVING EQUATIONS BY RADICALSGalois' criterionComposition series, Jordan-Holder theoremSolving cubics by radicals Worked examplesEIGENVECTORS, SPECTRAL THEOREMSEigenvectors, eigenvalues Diagonalizability, semi-simplicityCommuting endomorphisms ST = TS Inner product spaces Projections without coordinates Unitary operators Spectral theorems Corollaries of the spectral theoremWorked examples DUALS, NATURALITY, BILINEAR FORMS Dual vector spaces First example of naturality Bilinear forms Worked examples DETERMINANTS I Prehistory Definitions Uniqueness and other properties Existence TENSOR PRODUCTS Desiderata Definitions, uniqueness, existence First examples Tensor products f × g of mapsExtension of scalars, functoriality, naturality Worked examplesEXTERIOR POWERS DesiderataDefinitions, uniqueness, existenceSome elementary facts Exterior powers ?nf of mapsExterior powers of free modules Determinants revisitedMinors of matrices Uniqueness in the structure theorem Cartan's lemmaWorked examples