ABSTRACT ALGEBRA WITH APPLICATIONS

ABSTRACTALGEBRA WITHAPPLICATIONSIN T W OVOLUMESVOLUTvlH I IRINGS AND FIELDSKARLHEINZ SPINDLERDarmstadt,Marcel Dekker, Inc.GermanyNew York«Basel»Hong Kong

30. Analytical methods in group theory644Tangent space of a matrix group 645/ One-parameter subgroups 647/ Lie object of amatrix group 648/ Lie algebras 650/ Examples 650/ Relation between a closed matrixgroup and its Lie algebra 654/ Adjoint action 655/ Baker-Campbell-Hausdorff formula660/ Analytic matrix groups and their Lie group topology 662/ Exercises 66631. Groups in topology671Basic ideas in algebraic topology 671/ Homotopy 673/ Homotopy groups 677/ Fixedpoint theorems 679/ Commutativity of the higher homotopy groups 682/ Homology683/ Homology groups 688/ H\(X) as the abelianization of iti(X) 689/ Knot groups693/ Wirtinger presentation 693/ Exercises 697Appendix701A. Sets and functions 701/ B. Relations 709/ C. Cardinal and ordinal numbers 718/D. Point-set topology 728/ E. Categories and functors 737Bibliography743Index745Volume IIPrefacevRINGS AND FIELDS1. Introduction: The art of doing arithmetic1Euler's and Fermat's theorem 2/ Divisibility rules 2/ Other examples for the use of congruence classes tö obtain number-theoretical results 3/ Wilson's theorem 4/ Exercises 62. Rings and ring homomorphisms10Rings, commutative rings and unital rings 10/ Subrings 11/ Examples 11/ Power seriesrings 12/ Polynomial rings 13/ Matrix rings 15/ Rings of functions 15/ Convolutionrings 16/ Direct products and sums 17/ Ring homomorphisms, isomorphisms and embeddings 17/ Exercises 203. Integral domains and fields29Zero-divisors 29/ Nilpotent elements 29/ Units 31/ Examples 31/ Divisibility 35/ Integral domains 36/ Fields and skew-fields 38/ Quotient fields 40/ Application: Mikusinski's Operator calculus 41/ Exercises 48Contents / xi

4. Polynomial and power series rings54Polynomials in one or more variables 54/ Division with remainder 5 7 / Roots and theirmultiplicities 5 8 / Derivatives 60/ Symmetrie polynomials 6 1 / Main theorem on Symmetrie polynomials 6 4 / Discriminant 6 5 / Exercises 675. Ideals and quotient rings76Ideals 76/ Ideals generated by a set 77/ Principal ideals 77/ Simple rings 7 8 / Quotientrings 79/ Isomorphism theorems 80/ Maximal ideals 82/ Chinese remainder theorem8 3 / Exercises 866. Ideals in commutative rings94Prime and primary ideals 94/ Ideal quotients 96/ Radical of an ideal; radical ideals 96/Primary decompositions 102/ Symbolic powers 105/ Exercises 1077. Factorization in integral domains112Prime and irreducible elements 112/ Expressing notions of divisibility in terms of ideals114/ Factorization domains and unique factorization domains 116/ Characterization ofprineipal ideal domains 119/ Euclidean domains 119/ Examples 120/ Euclidean algorithm 125/ Exercises 1288. Factorization in polynomial and power series rings135Transmission of the unique factorization property from R to R[xi,.,xn] 135/ Uniquefactorization property for power series rings 139/ Factorization algorithms for polynomials 142/ Irreducibility criteria for polynomials 147/ Resultant of two polynomials150/ Decomposition of homogeneous polynomials 154/ Exercises 1559. Number-theoretical applications of unique factorization. . .163Representability of prime numbers by quadratic forms 163/ Legendre symbol 164/Quadratic reeiprocity law 166/ Three theorems of Fermat 169/ Ramanujan-Nagell theorem 172/ Insolvability of the equation x3 y3 z3 in N 174/ Kummer's theorem 178/Exercises 18010. Modules and integral ring extensions185Modules 185/ Free modules 185/ Submodules and quotient modules 186/ Module homomorphisms 187/ Noetherian and Artinian modules 187/ Algebras over commutativerings 1 9 1 / Algebraic and integral ring extensions 191/ Module-theoretical characterization of integral elements 193/ Integral closure 194/ Exercises 19611. Noetherian rings203Characterization of Noetherian rings 203/ Examples 203/ Hubert's Basis Theorem 205/Cohen's theorem 205/ Noetherian induetion 208/ Primary decompositions in Noetherian rings 2 0 9 / Artinian rings 212/ Exercises 215xii / Contents

12. Field extensions217Field extensions 217/ Intermediate fields 217/ Minimal polynomial 219/ Degree of simple extensions 220/ Degree formula 2 2 1 / Lüroth's theorem 223/ Cardinality of finitefields 224/ Trace and norm of finite field extensions 225/ Aigebraic extensions 226/Chaxacterization of simple extensions 227/ Algebraically generating and independentsets 228/ Transcendence bases 229/ Transcendence degree 2 3 1 / Decomposing a fieldextension into an aigebraic and a purely transcendental extension 232/ Zariski's lemma232/ Ruler and compass constructions 233/ Exercises 23813. Splitting fields and normal extensions248Adjoining roots of polynomials 248/ Splitting fields 250/ Normal extensions 250/ Extensions of field isomorpbisms 250/ Aigebraic closures 253/ Conjugates of an aigebraicelement 256/ Normal closures 257/ Finite fields 259/ Exercises 26214. Separability of field extensions268Separability 268/ Perfect fields 269/ Existence of primitive elements 2 7 1 / Characterization of separability via embeddings 272/ Properties of separable extensions 273/Pure inseparability 275/ Decomposition of an aigebraic field extension into a separableand a purely inseparable extension 277/ Degree of separability and inseparability 278/Trace and norm 279/ Discriminant of a finite field extension 283/ Chaxacterization ofseparability via the trace 285/ Exercises 28715. Field theory and integral ring extensions291Trace, norm and minimal polynomial of an integral element 292/ Integral closure ofZ in quadratic number fields 293/ Existence of integral bases 295/ Discriminant of anaigebraic number field 296/ Stickelberger's theorem 296/ Integral closure of Z in specialcyclotomic fields 297/ Exercises 29916. Affine algebras301Affine algebras 3 0 1 / Noether's normalization theorem 302/ Zariski's lemma 304/Stronger versions of Noether's normalization theorem 306/ Krull dimension of a commutative ring 308/ Lying-over theorem 309/ Going-up theorem 310/ Going-down theorem312/ Krull dimension of an affine algebra 313/ Exercises 31517. Ring theory and aigebraic geometry318Affine varieties 318/ Correspondence between varieties and ideals 322/ Hilbert's Nullstellensatz 324/ Zariski topology 326/ Irreducible varieties 327/ Polynomial and rational mappings between varieties 330/ Exercises 34218. Localization347Tangent space of a variety at a point 348/ Regulär and singular points 3 5 1 / Coincidence of the geometric and the aigebraic dimension of a variety 352/ Local ring of avariety at a point 353/ Localization of a commutative ring at a prime ideal 355/ Localrings 360/ Exercises 361Contents / xiii

19. Factorization of ideals365Kummer's idea to overcome non-unique factorization 365/ Dedekind domains 367/Fractional ideals 367/ Calculus of fractional ideals 368/ Characterization of Dedekinddomains 373/ Examples of Dedekind domains 373/ Divisibility of ideals in a Dedekinddomain 374/ Exercises 37720. Introduction to Galois theory: Solving polynomial equations379Quadratic formula 379/ Cardano's formula for cubic equations 380/ Ferrari's formulafor quartic equations 382/ Exercises 38421. The Galois group of a field extension388Galois group of a field extension 388/ Examples 388/ Fixed fields 390/ Correspondencebetween intermediate fields of (L : K) and subgroups of ?J 391/ Quantitative aspectsof the Galois correspondence 394/ Dedekind's theorem 395/ Artin's theorem 397/ Galois extensions 399/ Exercises 40122. Algebraic Galois extensions404Characterization of algebraic Galois extensions 404/ Separable and inseparable closurein a normal field extension 405/ Main theorem of Galois theory 407/ Examples 408/Galois groups of finite fields 413/ Primitive element theorem 414/ Constructions with aruler and a compass 415/ Normal bases 416/ Compositum of Galois extensions 419/ Infinite algebraic Galois extensions 420/ Topological characterization of closed subgroupsof a Galois group 425/ Exercises 42923. The Galois group of a polynomial435The Galois group of a polynomial as a group of permutations of its roots 435/Algorithm to determine the Galois group of a polynomial 436/ Reducing coefncients moduloa prime ideal 437/ Correspondence between the Galois group of a polynomial and theGalois group of a field extension 438/ Characterization of the irreducibility of a polynomial by the transitivity of its Galois group 439/ Dedekind's reciprocity theorem 441/Galois groups of cubic and quartic polynomials 443/ Galois groups of certain polynomials of prime degree 445/ Dedekind's theorem on reduction modulo a prime 446/ Newproof of the main theorem on Symmetrie polynomials 448/ Galois group of the generalpolynomial 449/ Exercises 45024. Roots of unity and cyclotomic polynomials455Roots of unity 455/ Cyclotomic polynomials 457/ Irreducibility of the cyclotomic polynomials over Q 460/ Galois group of xn — 1 460/ Examples 461/ Gaussian periods463/ Kummer's lemma 465/ Wedderburn's theorem 467/ Constructibility of regulärpolygons 468/ New proof of the quadratic reciprocity law 470/ Exercises 47225. Pure equations and cyclic extensions476Pure polynomials yield cyclic Galois groups 476/ Theorem 90 of Hubert 477/ CyclicGalois groups yield pure polynomials 478/ Lagrange resolvents 479/ Artin-Schreiertheorem 480/ Irreducibility of pure polynomials 481/ Kummer correspondence 484/Exercises 487xiv / Contents

26. Solvable equations and radical extensions490Solvability by radicals 490/ Radical extensions and their properties 490/ Strong radicalextensions 493/ Role of the roots of unity 494/ Expressing the solvability of a polynomial equation by the solvability of its Galois group 496/ Examples 499/ Algorithm todetermine the solvability of a given polynomial equation 501/ Examples: Quadratic,cubic and quartic equations 503/ Exercises 51227. Epilogue: The idea of Lie theory as a Galois theory for tents / xv

7. Factorization in integral domains 112 Prime and irreducible elements 112/ Expressing notions of divisibility in terms of ideals 114/ Factorization domains and unique factorization domains 116/ Characterization of prineipal ideal domains 119/ Euclidean domains 119/