PREFACE INTRODUCTION THE INTEGERS Unique factorization Irrationalities Z/m, the integers mod m Fermat's little theorem Sun-Ze's theorem Worked examples GROUPS I Groups Subgroups Homomorphisms, kernels, normal subgroups Cyclic groups Quotient groups Groups acting on sets The Sylow theorem Trying to classify finite groups, part I Worked examples THE PLAYERS: RINGS, FIELDS Rings, fields Ring homomorphisms Vector spaces, modules, algebras Polynomial rings I COMMUTATIVE RINGS I Divisibility and ideals Polynomials in one variable over a field Ideals and quotients Ideals and quotient rings Maximal ideals and fields Prime ideals and integral domains Fermat-Euler on sums of two squares Worked examples LINEAR ALGEBRA I: DIMENSION Some 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 examples CYCLOTOMIC POLYNOMIALS Multiple factors in polynomials Cyclotomic polynomials Examples Finite subgroups of fields Infinitude of primes p = 1 mod n Worked examples FINITE FIELDS Uniqueness Frobenius automorphisms Counting irreducibles MODULES OVER PIDS The structure theorem Variations Finitely generated abelian groups Jordan canonical form Conjugacy versus k[x]-module isomorphism Worked examples FINITELY GENERATED MODULES Free modules Finitely 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 THEORY Sets Posets, ordinals Transfinite induction Finiteness, infiniteness Comparison of infinities Example: transfinite Lagrange replacement Equivalents of the axiom of choice SYMMETRIC POLYNOMIALS The theorem First examples A variant: discriminants EISENSTEIN'S CRITERION Eisenstein's irreducibility criterion Examples VANDERMONDE DETERMINANTS Vandermonde determinants Worked examples CYCLOTOMIC POLYNOMIALS II Cyclotomic polynomials over Z Worked examples ROOTS OF UNITY Another proof of cyclicness Roots of unity Q with roots of unity adjoined Solution in radicals, Lagrange resolvents Quadratic fields, quadratic reciprocity Worked examples CYCLOTOMIC III Prime-power cyclotomic polynomials over Q Irreducibility of cyclotomic polynomials over Q Factoring Fn(x) in Fp[x] with p|n Worked examples PRIMES IN ARITHMETIC PROGRESSIONS Euler's theorem and the zeta function Dirichlet's theorem Dual 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 extensions Primitive elements Normal field extensions The main theorem Conjugates, trace, norm Basic examples Worked examples SOLVING EQUATIONS BY RADICALS Galois' criterion Composition series, Jordan-Holder theorem Solving cubics by radicals Worked examples EIGENVECTORS, SPECTRAL THEOREMS Eigenvectors, eigenvalues Diagonalizability, semi-simplicity Commuting endomorphisms ST = TS Inner product spaces Projections without coordinates Unitary operators Spectral theorems Corollaries of the spectral theorem Worked 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 x g of maps Extension of scalars, functoriality, naturality Worked examples EXTERIOR POWERS Desiderata Definitions, uniqueness, existence Some elementary facts Exterior powers ?nf of maps Exterior powers of free modules Determinants revisited Minors of matrices Uniqueness in the structure theorem Cartan's lemma Worked examples

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