Basic algebra / Nathan Jacobson.
Series Dover books on mathematicsEditor: Mineola, N.Y. : Dover Publications, 2009Edición: 2nd ed., Dover edDescripción: 2 v. : il. ; 24 cmISBN: 9780486471891 (v. 1 : pbk.); 0486471896 (v. 1 : pbk.); 9780486471877 (v. 2 : pbk.); 048647187X (v. 2 : pbk.)Tema(s): AlgebraOtra clasificación: 00A05 (12-01 15-01 16-01)Vol. I Preface xi Preface to the First Edition xiii INTRODUCTION: CONCEPTS FROM SET THEORY. THE INTEGERS [1] 0.1 The power set of a set [3] 0.2 The Cartesian product set. Maps [4] 0.3 Equivalence relations. Factoring a map through an equivalence relation [10] 0.4 The natural numbers [15] 0.5 The number system Z of integers [19] 0.6 Some basic arithmetic facts about Z [22] 0.7 A word on cardinal numbers [24] 1 MONOIDS AND GROUPS [26] 1.1 Monoids of transformations and abstract monoids [28] 1.2 Groups of transformations and abstract groups [31] 1.3 Isomorphism. Cayley’s theorem [36] 1.4 Generalized associativity. Commutativity [39] 1.5 Submonoids and subgroups generated by a subset. Cyclic groups [42] 1.6 Cycle decomposition of permutations [48] 1.7 Orbits. Cosets of a subgroup [51] 1.8 Congruences. Quotient monoids and groups [54] 1.9 Homomorphisms [58] 1.10 Subgroups of a homomorphic image. Two basic isomorphism theorems [64] 1.11 Free objects. Generators and relations [67] 1.12 Groups acting on sets [71] 1.13 Sylow’s theorems [79] RINGS [85] 2.1 Definition and elementary properties [86] 2.2 Types of rings [90] 2.3 Matrix rings [92] 2.4 Quaternions [98] 2.5 Ideals, quotient rings [101] 2.6 Ideals and quotient rings for Z [103] 2.7 Homomorphisms of rings. Basic theorems [106] 2.8 Anti-isomorphisms [111] 2.9 Field of fractions of a commutative domain [115] 2.10 Polynomial rings [119] 2.11 Some properties of polynomial rings and applications [127] 2.12 Polynomial functions [134] 2.13 Symmetric polynomials [138] 2.14 Factorial monoids and rings [140] 2.15 Principal ideal domains and Euclidean domains [147] 2.16 Polynomial extensions of factorial domains [151] 2.17 “Rngs” (rings without unit) [155] 3 MODULES OVER A PRINCIPAL IDEAL DOMAIN [157] 3.1 Ring of endomorphisms of an abelian group [158] 3.2 Left and right modules [163] 3.3 Fundamental concepts and results [166] 3.4 Free modules and matrices [170] 3.5 Direct sums of modules [175] 3.6 Finitely generated modules over a p.i.d. Preliminary results [179] 3.7 Equivalence of matrices with entries in a p.i.d. [181] 3.8 Structure theorem for finitely generated modules over a p.i.d. [187] 3.9 Torsion modules, primary components, invariance theorem [189] 3.10 Applications to abelian groups and to linear transformations [194] 3.11 The ring of endomorphisms of a finitely generated module over a p.i.d. [204] 4 GALOIS THEORY OF EQUATIONS [210] 4.1 Preliminary results, some old, some new [213] 4.2 Construction with straight-edge and compass [216] 4.3 Splitting field of a polynomial [224] 4.4 Multiple roots [229] 4.5 The Galois group. The fundamental Galois pairing [234] 4.6 Some results on finite groups [244] 4.7 Galois’ criterion for solvability by radicals [251] 4.8 The Galois group as permutation group of the roots [256] 4.9 The general equation of the nth degree [262] 4.10 Equations with rational coefficients and symmetric group as Galois group [267] 4.11 Constructible regular n-gons [271] 4.12 Transcendence of e and π. The Lindemann-Weierstrass theorem [277] 4.13 Finite fields [287] 4.14 Special bases for finite dimensional extensions fields [290] 4.15 Traces and norms [296] 4.16 Mod p reduction [301] 5 REAL POLYNOMIAL EQUATIONS AND INEQUALITIES [306] 5.1 Ordered fields. Real closed fields [307] 5.2 Sturm’s theorem [311] 5.3 Formalized Euclidean algorithm and Sturm’s theorem [316] 5.4 Elimination procedures. Resultants [322] 5.5 Decision method for an algebraic curve [327] 5.6 Tarski’s theorem [335] 6 METRIC VECTOR SPACES AND THE CLASSICAL GROUPS [342] 6.1 Linear functions and bilinear forms [343] 6.2 Alternate forms [349] 6.3 Quadratic forms and symmetric bilinear forms [354] 6.4 Basic concepts of orthogonal geometry [361] 6.5 Witt’s cancellation theorem [367] 6.6 The theorem of Cartan-Dieudonné [371] 6.7 Structure of the general linear group GLn(F) [375] 6.8 Structure of orthogonal groups [382] 6.9 Symplectic geometry. The symplectic group [391] 6.10 Orders of orthogonal and symplectic groups over a finite field [398] 6.11 Postscript on hermitian forms and unitary geometry [401] 7 ALGEBRAS OVER A FIELD [405] 7.1 Definition and examples of associative algebras [406] 12 Exterior algebras. Application to determinants [411] 7.3 Regular matrix representations of associative algebras. Norms and traces [422] 7.4 Change of base field. Transitivity of trace and norm [426] 7.5 Non-associative algebras. Lie and Jordan algebras [430] 7.6 Hurwitz’ problem. Composition algebras [438] 7.7 Frobenius’ and Wedderburn’s theorems on associative division algebras [451] 8 LATTICES AND BOOLEAN ALGEBRAS [455] 8.1 Partially ordered sets and lattices [456] 8.2 Distributivity and modularity [461] 8.3 The theorem of Jordan-Hölder-Dedekind [466] 8.4 The lattice of subspaces of a vector space. Fundamental theorem of projective geometry [468] 8.5 Boolean algebras [474] 8.6 The Mobius function of a partially ordered set [480] Appendix [489] Index [493]
Vol. II Contents of Basic Algebra I x Preface xiii Preface to the First Edition xv INTRODUCTION [1] 0.1 Zorn's lemma [2] 0.2 Arithmetic of cardinal numbers [3] 0.3 Ordinal and cardinal numbers [4] 0.4 Sets and classes [6] References [7] 1 CATEGORIES [8] 1.1 Definition and examples of categories [9] 1.2 Some basic categorical concepts [15] 1.3 Functors and natural transformations [18] 1.4 Equivalence of categories [26] 1.5 Products and coproducts [32] 1.6 The horn functors. Representable functors [37] 1.7 Universals [40] 1.8 Adjoints [45] References [51] 2 UNIVERSAL ALGEBRA [52] 2.1 fl-algebras [53] 2.2 Subalgebras and products [58] 2.3 Homomorphisms and congruences [60] 2.4 The lattice of congruences. Subdirect products [66] 2.5 Direct and inverse limits [70] 2.6 Ultraproducts [75] 2.7 Free Ω-algebras [78] 2.8 Varieties [81] 2.9 Free products of groups [87] 2.10 Internal characterization of varieties [91] References [93] 3 MODULES [94] 3.1 The categories R-mod and mod-R [95] 3.2 Artinian and Noetherian modules [100] 3.3 Schreier refinement theorem. Jordan-Hölder theorem [104] 3.4 The Krull-Schmidt theorem [110] 3.5 Completely reducible modules [117] 3.6 Abstract dependence relations. Invariance of dimensionality [122] 3.7 Tensor products of modules [125] 3.8 Bimodules [133] 3.9 Algebras and coalgebras [137] 3.10 Projective modules [148] 3.11 Injective modules. Injective hull [156] 3.12 Morita contexts [164] 3.13 The Wedderburn-Artin theorem for simple rings [171] 3.14 Generators and progenerators [173] 3.15 Equivalence of categories of modules [177] References [183] 4 BASIC STRUCTURE THEORY OF RINGS [184] 4.1 Primitivity and semi-primitivity [185] 4.2 The radical of a ring [192] 4.3 Density theorems [197] 4.4 Artinian rings [202] 4.5 Structure theory of algebras [210] 4.6 Finite dimensional central simple algebras [215] 4.7 The Brauer group [226] 4.8 Clifford algebras [228] References [245] 5 CLASSICAL REPRESENTATION THEORY OF FINITE GROUPS [246] 5.1 Representations and matrix representations of groups [247] 5.2 Complete reducibility [251] 5.3 Application of the representation theory of algebras [257] 5.4 Irreducible representations of Sn [265] 5.5 Characters. Orthogonality relations [269] 5.6 Direct products of groups. Characters of abelian groups [279] 5.7 Some arithmetical considerations [282] 5.8 Burnside’s pa qb theorem [284] 5.9 Induced modules [286] 5.10 Properties of induction. Frobenius reciprocity theorem [292] 5.11 Further results on induced modules [299] 5.12 Brauer’s theorem on induced characters [305] 5.13 Brauer’s theorem on splitting fields [313] 5.14 The Schur index [314] 5.15 Frobenius groups [317] References [325] 6 ELEMENTS OF HOMOLOGICAL ALGEBRA WITH APPLICATIONS [326] 6.1 Additive and abelian categories [327] 6.2 Complexes and homology [331] 6.3 Long exact homology sequence [334] 6.4 Homotopy [337] 6.5 Resolutions [339] 6.6 Derived functors [342] 6.7 Ext [346] 6.8 Tor [353] 6.9 Cohomology of groups [355] 6.10 Extensions of groups [363] 6.11 Cohomology of algebras [370] 6.12 Homological dimension [375] 6.13 Koszul’s complex and Hilbert’s syzygy theorem 378 References [387] 7 COMMUTATIVE IDEAL THEORY: GENERAL THEORY AND NOETHERIAN RINGS [388] 7.1 Prime ideals. Nil radical [389] 7.2 Localization of rings [393] 7.3 Localization of modules [397] 7.4 Localization at the complement of a prime ideal. Local-global relations [400] 7.5 Prime spectrum of a commutative ring [403] 7.6 Integral dependence [408] 7.7 Integrally closed domains [412] 7.8 Rank of projective modules [414] 7.9 Projective class group [419] 7.10 Noetherian rings [420] 7.11 Commutative artinian rings [425] 7.12 Affine algebraic varieties. The Hilbert Nullstellensatz [427] 7.13 Primary decompositions [433] 7.14 Artin-Rees lemma. Krull intersection theorem [440] 7.15 Hilbert’s polynomial for a graded module [443] 7.16 The characteristic polynomial of a noetherian local ring [448] 7.17 Krull dimension [450] 7.18 I-adic topologies and completions 455 References [462] FIELD THEORY [463] 8.1 Algebraic closure of a field [464] 8.2 The Jacobson-Bourbaki correspondence [468] 8.3 Finite Galois theory [471] 8.4 Crossed products and the Brauer group [475] 8.5 Cyclic algebras [484] 8.6 Infinite Galois theory [486] 8.7 Separability and normality [489] 8.8 Separable splitting fields [495] 8.9 Kummer extensions [498] 8.10 Rings of Witt vectors [501] 8.11 Abelian p-extension [509] 8.12 Transcendency bases [514] 8.13 Transcendency bases for domains. Affine algebras [517] 8.14 Luroth’s theorem [520] 8.15 Separability for arbitrary extension fields [525] 8.16 Derivations [530] 8.17 Galois theory for purely inseparable extensions of exponent one [541] 8.18 Tensor products of fields [544] 8.19 Free composites of fields [550] References [556] 9 VALUATION THEORY [557] 9.1 Absolute values [558] 9.2 The approximation theorem [562] 9.3 Absolute values on Q and F(x) [564] 9.4 Completion of a field [566] 9.5 Finite dimensional extensions of complete fields. The archimedean case [569] 9.6 Valuations [573] 9.7 Valuation rings and places [577] 9.8 Extension of homomorphisms and valuations [580] 9.9 Determination of the absolute values of a finite dimensional extension field [585] 9.10 Ramification index and residue degree. Discrete valuations [588] 9.11 Hensel’s lemma [592] 9.12 Local fields [595] 9.13 Totally disconnected locally compact division rings [599] 9.14 The Brauer group of a local field [608] 9.15 Quadratic forms over local fields 611 References [618] 10 DEDEKIND DOMAINS [619] 10.1 Fractional ideals. Dedekind domains [620] 10.2 Characterizations of Dedekind domains [625] 10.3 Integral extensions of Dedekind domains [631] 10.4 Connections with valuation theory [634] 10.5 Ramified primes and the discriminant [639] 10.6 Finitely generated modules over a Dedekind domain 643 References [649] 11 FORMALLY REAL FIELDS [650] 11.1 Formally real fields [651] 11.2 Real closures [655] 11.3 Totally positive elements [657] 11.4 Hilbert’s seventeenth problem [660] 11.5 Pfister theory of quadratic forms [663] 11.6 Sums of squares in R(x1..., xn), R a real closed field [669] 11.7 Artin-Schreier characterization of real closed fields 674 References [677] INDEX [679]
| Item type | Home library | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | 00A05A J17-2 (Browse shelf) | Vol. I | Available | A-8802 | ||
|
Libros
|
Instituto de Matemática, CONICET-UNS | 00A05A J17-2 (Browse shelf) | Vol. II | Available | A-8803 |
Originally published: 2nd ed. San Francisco : W.H. Freeman, 1985-1989.
Vol. I
Preface xi --
Preface to the First Edition xiii --
INTRODUCTION: CONCEPTS FROM SET THEORY. THE INTEGERS [1] --
0.1 The power set of a set [3] --
0.2 The Cartesian product set. Maps [4] --
0.3 Equivalence relations. Factoring a map through an equivalence relation [10] --
0.4 The natural numbers [15] --
0.5 The number system Z of integers [19] --
0.6 Some basic arithmetic facts about Z [22] --
0.7 A word on cardinal numbers [24] --
1 MONOIDS AND GROUPS [26] --
1.1 Monoids of transformations and abstract monoids [28] --
1.2 Groups of transformations and abstract groups [31] --
1.3 Isomorphism. Cayley’s theorem [36] --
1.4 Generalized associativity. Commutativity [39] --
1.5 Submonoids and subgroups generated by a subset. Cyclic groups [42] --
1.6 Cycle decomposition of permutations [48] --
1.7 Orbits. Cosets of a subgroup [51] --
1.8 Congruences. Quotient monoids and groups [54] --
1.9 Homomorphisms [58] --
1.10 Subgroups of a homomorphic image. Two basic isomorphism theorems [64] --
1.11 Free objects. Generators and relations [67] --
1.12 Groups acting on sets [71] --
1.13 Sylow’s theorems [79] --
RINGS [85] --
2.1 Definition and elementary properties [86] --
2.2 Types of rings [90] --
2.3 Matrix rings [92] --
2.4 Quaternions [98] --
2.5 Ideals, quotient rings [101] --
2.6 Ideals and quotient rings for Z [103] --
2.7 Homomorphisms of rings. Basic theorems [106] --
2.8 Anti-isomorphisms [111] --
2.9 Field of fractions of a commutative domain [115] --
2.10 Polynomial rings [119] --
2.11 Some properties of polynomial rings and applications [127] --
2.12 Polynomial functions [134] --
2.13 Symmetric polynomials [138] --
2.14 Factorial monoids and rings [140] --
2.15 Principal ideal domains and Euclidean domains [147] --
2.16 Polynomial extensions of factorial domains [151] --
2.17 “Rngs” (rings without unit) [155] --
3 MODULES OVER A PRINCIPAL IDEAL DOMAIN [157] --
3.1 Ring of endomorphisms of an abelian group [158] --
3.2 Left and right modules [163] --
3.3 Fundamental concepts and results [166] --
3.4 Free modules and matrices [170] --
3.5 Direct sums of modules [175] --
3.6 Finitely generated modules over a p.i.d. Preliminary results [179] --
3.7 Equivalence of matrices with entries in a p.i.d. [181] --
3.8 Structure theorem for finitely generated modules over a p.i.d. [187] --
3.9 Torsion modules, primary components, invariance theorem [189] --
3.10 Applications to abelian groups and to linear transformations [194] --
3.11 The ring of endomorphisms of a finitely generated module over a p.i.d. [204] --
4 GALOIS THEORY OF EQUATIONS [210] --
4.1 Preliminary results, some old, some new [213] --
4.2 Construction with straight-edge and compass [216] --
4.3 Splitting field of a polynomial [224] --
4.4 Multiple roots [229] --
4.5 The Galois group. The fundamental Galois pairing [234] --
4.6 Some results on finite groups [244] --
4.7 Galois’ criterion for solvability by radicals [251] --
4.8 The Galois group as permutation group of the roots [256] --
4.9 The general equation of the nth degree [262] --
4.10 Equations with rational coefficients and symmetric group as Galois group [267] --
4.11 Constructible regular n-gons [271] --
4.12 Transcendence of e and π. The Lindemann-Weierstrass theorem [277] --
4.13 Finite fields [287] --
4.14 Special bases for finite dimensional extensions fields [290] --
4.15 Traces and norms [296] --
4.16 Mod p reduction [301] --
5 REAL POLYNOMIAL EQUATIONS AND INEQUALITIES [306] --
5.1 Ordered fields. Real closed fields [307] --
5.2 Sturm’s theorem [311] --
5.3 Formalized Euclidean algorithm and Sturm’s theorem [316] --
5.4 Elimination procedures. Resultants [322] --
5.5 Decision method for an algebraic curve [327] --
5.6 Tarski’s theorem [335] --
6 METRIC VECTOR SPACES AND THE CLASSICAL GROUPS [342] --
6.1 Linear functions and bilinear forms [343] --
6.2 Alternate forms [349] --
6.3 Quadratic forms and symmetric bilinear forms [354] --
6.4 Basic concepts of orthogonal geometry [361] --
6.5 Witt’s cancellation theorem [367] --
6.6 The theorem of Cartan-Dieudonné [371] --
6.7 Structure of the general linear group GLn(F) [375] --
6.8 Structure of orthogonal groups [382] --
6.9 Symplectic geometry. The symplectic group [391] --
6.10 Orders of orthogonal and symplectic groups over a finite field [398] --
6.11 Postscript on hermitian forms and unitary geometry [401] --
7 ALGEBRAS OVER A FIELD [405] --
7.1 Definition and examples of associative algebras [406] --
12 Exterior algebras. Application to determinants [411] --
7.3 Regular matrix representations of associative algebras. Norms and traces [422] --
7.4 Change of base field. Transitivity of trace and norm [426] --
7.5 Non-associative algebras. Lie and Jordan algebras [430] --
7.6 Hurwitz’ problem. Composition algebras [438] --
7.7 Frobenius’ and Wedderburn’s theorems on associative division algebras [451] --
8 LATTICES AND BOOLEAN ALGEBRAS [455] --
8.1 Partially ordered sets and lattices [456] --
8.2 Distributivity and modularity [461] --
8.3 The theorem of Jordan-Hölder-Dedekind [466] --
8.4 The lattice of subspaces of a vector space. Fundamental theorem of projective geometry [468] --
8.5 Boolean algebras [474] --
8.6 The Mobius function of a partially ordered set [480] --
Appendix [489] --
Index [493] --
Vol. II
Contents of Basic Algebra I x --
Preface xiii --
Preface to the First Edition xv --
INTRODUCTION [1] --
0.1 Zorn's lemma [2] --
0.2 Arithmetic of cardinal numbers [3] --
0.3 Ordinal and cardinal numbers [4] --
0.4 Sets and classes [6] --
References [7] --
1 CATEGORIES [8] --
1.1 Definition and examples of categories [9] --
1.2 Some basic categorical concepts [15] --
1.3 Functors and natural transformations [18] --
1.4 Equivalence of categories [26] --
1.5 Products and coproducts [32] --
1.6 The horn functors. Representable functors [37] --
1.7 Universals [40] --
1.8 Adjoints [45] --
References [51] --
2 UNIVERSAL ALGEBRA [52] --
2.1 fl-algebras [53] --
2.2 Subalgebras and products [58] --
2.3 Homomorphisms and congruences [60] --
2.4 The lattice of congruences. Subdirect products [66] --
2.5 Direct and inverse limits [70] --
2.6 Ultraproducts [75] --
2.7 Free Ω-algebras [78] --
2.8 Varieties [81] --
2.9 Free products of groups [87] --
2.10 Internal characterization of varieties [91] --
References [93] --
3 MODULES [94] --
3.1 The categories R-mod and mod-R [95] --
3.2 Artinian and Noetherian modules [100] --
3.3 Schreier refinement theorem. Jordan-Hölder theorem [104] --
3.4 The Krull-Schmidt theorem [110] --
3.5 Completely reducible modules [117] --
3.6 Abstract dependence relations. Invariance of dimensionality [122] --
3.7 Tensor products of modules [125] --
3.8 Bimodules [133] --
3.9 Algebras and coalgebras [137] --
3.10 Projective modules [148] --
3.11 Injective modules. Injective hull [156] --
3.12 Morita contexts [164] --
3.13 The Wedderburn-Artin theorem for simple rings [171] --
3.14 Generators and progenerators [173] --
3.15 Equivalence of categories of modules [177] --
References [183] --
4 BASIC STRUCTURE THEORY OF RINGS [184] --
4.1 Primitivity and semi-primitivity [185] --
4.2 The radical of a ring [192] --
4.3 Density theorems [197] --
4.4 Artinian rings [202] --
4.5 Structure theory of algebras [210] --
4.6 Finite dimensional central simple algebras [215] --
4.7 The Brauer group [226] --
4.8 Clifford algebras [228] --
References [245] --
5 CLASSICAL REPRESENTATION THEORY OF FINITE GROUPS [246] --
5.1 Representations and matrix representations of groups [247] --
5.2 Complete reducibility [251] --
5.3 Application of the representation theory of algebras [257] --
5.4 Irreducible representations of Sn [265] --
5.5 Characters. Orthogonality relations [269] --
5.6 Direct products of groups. Characters of abelian groups [279] --
5.7 Some arithmetical considerations [282] --
5.8 Burnside’s pa qb theorem [284] --
5.9 Induced modules [286] --
5.10 Properties of induction. Frobenius reciprocity theorem [292] --
5.11 Further results on induced modules [299] --
5.12 Brauer’s theorem on induced characters [305] --
5.13 Brauer’s theorem on splitting fields [313] --
5.14 The Schur index [314] --
5.15 Frobenius groups [317] --
References [325] --
6 ELEMENTS OF HOMOLOGICAL ALGEBRA --
WITH APPLICATIONS [326] --
6.1 Additive and abelian categories [327] --
6.2 Complexes and homology [331] --
6.3 Long exact homology sequence [334] --
6.4 Homotopy [337] --
6.5 Resolutions [339] --
6.6 Derived functors [342] --
6.7 Ext [346] --
6.8 Tor [353] --
6.9 Cohomology of groups [355] --
6.10 Extensions of groups [363] --
6.11 Cohomology of algebras [370] --
6.12 Homological dimension [375] --
6.13 Koszul’s complex and Hilbert’s syzygy theorem 378 References [387] --
7 COMMUTATIVE IDEAL THEORY: GENERAL THEORY AND NOETHERIAN RINGS [388] --
7.1 Prime ideals. Nil radical [389] --
7.2 Localization of rings [393] --
7.3 Localization of modules [397] --
7.4 Localization at the complement of a prime ideal. Local-global relations [400] --
7.5 Prime spectrum of a commutative ring [403] --
7.6 Integral dependence [408] --
7.7 Integrally closed domains [412] --
7.8 Rank of projective modules [414] --
7.9 Projective class group [419] --
7.10 Noetherian rings [420] --
7.11 Commutative artinian rings [425] --
7.12 Affine algebraic varieties. The Hilbert Nullstellensatz [427] --
7.13 Primary decompositions [433] --
7.14 Artin-Rees lemma. Krull intersection theorem [440] --
7.15 Hilbert’s polynomial for a graded module [443] --
7.16 The characteristic polynomial of a noetherian local ring [448] --
7.17 Krull dimension [450] --
7.18 I-adic topologies and completions 455 References [462] --
FIELD THEORY [463] --
8.1 Algebraic closure of a field [464] --
8.2 The Jacobson-Bourbaki correspondence [468] --
8.3 Finite Galois theory [471] --
8.4 Crossed products and the Brauer group [475] --
8.5 Cyclic algebras [484] --
8.6 Infinite Galois theory [486] --
8.7 Separability and normality [489] --
8.8 Separable splitting fields [495] --
8.9 Kummer extensions [498] --
8.10 Rings of Witt vectors [501] --
8.11 Abelian p-extension [509] --
8.12 Transcendency bases [514] --
8.13 Transcendency bases for domains. Affine algebras [517] --
8.14 Luroth’s theorem [520] --
8.15 Separability for arbitrary extension fields [525] --
8.16 Derivations [530] --
8.17 Galois theory for purely inseparable extensions of exponent one [541] --
8.18 Tensor products of fields [544] --
8.19 Free composites of fields [550] --
References [556] --
9 VALUATION THEORY [557] --
9.1 Absolute values [558] --
9.2 The approximation theorem [562] --
9.3 Absolute values on Q and F(x) [564] --
9.4 Completion of a field [566] --
9.5 Finite dimensional extensions of complete fields. The archimedean case [569] --
9.6 Valuations [573] --
9.7 Valuation rings and places [577] --
9.8 Extension of homomorphisms and valuations [580] --
9.9 Determination of the absolute values of a finite dimensional extension field [585] --
9.10 Ramification index and residue degree. Discrete valuations [588] --
9.11 Hensel’s lemma [592] --
9.12 Local fields [595] --
9.13 Totally disconnected locally compact division rings [599] --
9.14 The Brauer group of a local field [608] --
9.15 Quadratic forms over local fields 611 References [618] --
10 DEDEKIND DOMAINS [619] --
10.1 Fractional ideals. Dedekind domains [620] --
10.2 Characterizations of Dedekind domains [625] --
10.3 Integral extensions of Dedekind domains [631] --
10.4 Connections with valuation theory [634] --
10.5 Ramified primes and the discriminant [639] --
10.6 Finitely generated modules over a Dedekind domain 643 References [649] --
11 FORMALLY REAL FIELDS [650] --
11.1 Formally real fields [651] --
11.2 Real closures [655] --
11.3 Totally positive elements [657] --
11.4 Hilbert’s seventeenth problem [660] --
11.5 Pfister theory of quadratic forms [663] --
11.6 Sums of squares in R(x1..., xn), R a real closed field [669] --
11.7 Artin-Schreier characterization of real closed fields 674 References [677] --
INDEX [679] --
MR, 86d:00001 (v. 1)
MR, 90m:00007 (v. 2)
Click on an image to view it in the image viewer
There are no comments on this title.