Abstract Algebra

Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal mapping properties, rather than by constructions whose technical details are irrelevant. Addresses Common Curricular Weaknesses In addition to standard introductory material on the subject, such as Lagrange's and Sylow's theorems in group theory, the text provides important specific illustrations of general theory, discussing in detail finite fields, cyclotomic polynomials, and cyclotomic fields. The book also focuses on broader background, including brief but representative discussions of naive set theory and equivalents of the axiom of choice, quadratic reciprocity, Dirichlet's theorem on primes in arithmetic progressions, and some basic complex analysis. Numerous worked examples and exercises throughout facilitate a thorough understanding of the material Cover Page......Page 1 Author's note......Page 2 Introduction......Page 3 Contents......Page 5 1.1 Unique factorization......Page 11 1.2 Irrationalities......Page 15 1.3 Z/m, the integers mod m......Page 16 1.4 Fermat’s Little Theorem......Page 18 1.5 Sun-Ze’s theorem......Page 21 1.6 Worked examples......Page 22 2.1 Groups......Page 27 2.2 Subgroups, Lagrange’s theorem......Page 29 2.3 Homomorphisms, kernels, normal subgroups......Page 32 2.4 Cyclic groups......Page 34 2.5 Quotient groups......Page 36 2.6 Groups acting on sets......Page 38 2.7 The Sylow theorem......Page 41 2.8 Trying to classify finite groups, part I......Page 44 2.9 Worked examples......Page 52 3.1 Rings, fields......Page 57 3.2 Ring homomorphisms......Page 60 3.3 Vectorspaces, modules, algebras......Page 62 3.4 Polynomial rings I......Page 64 4.1 Divisibility and ideals......Page 71 4.2 Polynomials in one variable over a field......Page 72 4.3 Ideals and quotients......Page 75 4.4 Ideals and quotient rings......Page 78 4.7 Fermat-Euler on sums of two squares......Page 79 4.8 Worked examples......Page 83 5.1 Some simple results......Page 89 5.2 Bases and dimension......Page 90 5.3 Homomorphisms and dimension......Page 92 6.1 Adjoining things......Page 95 6.2 Fields of fractions, fields of rational functions......Page 98 6.3 Characteristics, finite fields......Page 100 6.4 Algebraic field extensions......Page 102 6.5 Algebraic closures......Page 106 7.1 Irreducibles over a finite field......Page 109 7.2 Worked examples......Page 112 8.1 Multiple factors in polynomials......Page 115 8.2 Cyclotomic polynomials......Page 117 8.3 Examples......Page 120 8.5 Infinitude of primes p = 1 mod n......Page 123 8.6 Worked examples......Page 124 9.1 Uniqueness......Page 129 9.2 Frobenius automorphisms......Page 130 9.3 Counting irreducibles......Page 133 10.1 The structure theorem......Page 135 10.2 Variations......Page 136 10.3 Finitely-generated abelian groups......Page 138 10.4 Jordan canonical form......Page 140 10.5 Conjugacy versus k[x]-module isomorphism......Page 144 10.6 Worked examples......Page 151 11.1 Free modules......Page 161 11.2 Finitely-generated modules over a domain......Page 165 11.3 PIDs are UFDs......Page 168 11.4 Structure theorem, again......Page 169 11.6 Submodules of free modules......Page 171 12.1 Gauss’ lemma......Page 175 12.2 Fields of fractions......Page 177 12.3 Worked examples......Page 179 13.1 Cycles, disjoint cycle decompositions......Page 185 13.3 Worked examples......Page 186 14.1 Sets......Page 191 14.2 Posets, ordinals......Page 193 14.3 Transfinite induction......Page 197 14.5 Comparison of infinities......Page 198 14.6 Example: transfinite Lagrange replacement......Page 200 14.7 Equivalents of the Axiom of Choice......Page 201 15.1 The theorem......Page 203 15.2 First examples......Page 204 15.3 A variant: discriminants......Page 206 16.1 Eisenstein’s irreducibility criterion......Page 209 16.2 Examples......Page 210 17.1 Vandermonde determinants......Page 213 17.2 Worked examples......Page 216 18.1 Cyclotomic polynomials over Z......Page 221 18.2 Worked examples......Page 223 19.1 Another proof of cyclicness......Page 229 19.3 Q with roots of unity adjoined......Page 230 19.4 Solution in radicals, Lagrange resolvents......Page 237 19.5 Quadratic fields, quadratic reciprocity......Page 240 19.6 Worked examples......Page 244 20.1 Prime-power cyclotomic polynomials over Q......Page 253 20.2 Irreducibility of cyclotomic polynomials over Q......Page 255 20.4 Worked examples......Page 256 21.1 Euler’s theorem and the zeta function......Page 271 21.2 Dirichlet’s theorem......Page 273 21.3 Dual groups of abelian groups......Page 276 21.4 Non-vanishing on Re(s) = 1......Page 278 21.5 Analytic continuations......Page 279 21.6 Dirichlet series with positive coefficients......Page 280 22 Galois theory......Page 283 22.1 Field extensions, imbeddings, automorphisms......Page 284 22.2 Separable field extensions......Page 285 22.3 Primitive elements......Page 287 22.4 Normal field extensions......Page 288 22.5 The main theorem......Page 290 22.7 Basic examples......Page 292 22.8 Worked examples......Page 293 23.1 Galois’ criterion......Page 304 23.3 Solving cubics by radicals......Page 306 23.4 Worked examples......Page 309 24.1 Eigenvectors, eigenvaluesv......Page 314 24.2 Diagonalizability, semi-simplicity......Page 317 24.3 Commuting endomorphisms ST = TS24.3 Commuting endomorphisms ST = TS......Page 319 24.4 Inner product spaces......Page 320 24.6 Unitary operators......Page 325 24.7 Spectral theorems......Page 326 24.8 Corollaries of the spectral theorem......Page 327 24.9 Worked examples......Page 329 25.1 Dual vectorspaces......Page 336 25.2 First example of naturality......Page 340 25.3 Bilinear forms......Page 341 25.4 Worked examples......Page 344 26.1 Prehistory......Page 352 26.2 Definitions......Page 354 26.3 Uniqueness and other properties......Page 355 26.4 Existence......Page 359 27.1 Desiderata......Page 362 27.2 Definitions, uniqueness, existence......Page 363 27.3 First examples......Page 367 27.4 Tensor products f ⊗ g of maps......Page 370 27.5 Extension of scalars, functoriality, naturality......Page 371 27.6 Worked examples......Page 374 28.1 Desiderata......Page 386 28.2 Definitions, uniqueness, existence......Page 387 28.3 Some elementary facts......Page 390 28.4 Exterior powers Λn(f) of maps......Page 391 28.5 Exterior powers of free modules......Page 392 28.6 Determinants revisited......Page 395 28.7 Minors of matrices......Page 396 28.8 Uniqueness in the structure theorem......Page 397 28.9 Cartan’s lemma......Page 398 28.10 Cayley-Hamilton theorem......Page 399 28.11 Worked examples......Page 401