Basic notions of algebra:
Gespeichert in:
| Vorheriger Titel: | Algebra |
|---|---|
| 1. Verfasser: | |
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1997
|
| Ausgabe: | 2. printing |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Auch als: Algebra ; 1 |
| Beschreibung: | 258 S. Ill., graph. Darst. |
| ISBN: | 3540612211 |
Internformat
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| 100 | 1 | |a Šafarevič, Igorʹ R. |d 1923-2017 |e Verfasser |0 (DE-588)119280337 |4 aut | |
| 245 | 1 | 0 | |a Basic notions of algebra |c I. R. Shafarevich |
| 250 | |a 2. printing | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1997 | |
| 300 | |a 258 S. |b Ill., graph. Darst. | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | I. R. SHAFAREVICH BASIC NOTIONS OF ALGEBRA WITH 45 FIGURES SPRINGER
BASIC NOTIONS OF ALGEBRA I.R. SHAFAREVICH TRANSLATED FROM THE RUSSIAN BY
M. REID CONTENTS PREFACE 4 § 1. WHAT IS ALGEBRA? 6 THE IDEA OF
COORDINATISATION. EXAMPLES: DICTIONARY OF QUANTUM MECHANICS AND
COORDINATISATION OF FINITE MODELS OF INCIDENCE AXIOMS AND PARALLELISM.
§2. FIELDS 11 FIELD AXIOMS, ISOMORPHISMS. FIELD OF RATIONAL FUNCTIONS IN
INDEPENDENT VARIABLES; FUNCTION FIELD OF A PLANE ALGEBRAIC CURVE. FIELD
OF LAURENT SERIES AND FORMAL LAURENT SERIES. § 3. COMMUTATIVE RINGS 17
RING AXIOMS; ZERODIVISORS AND INTEGRAL DOMAINS. FIELD OF FRACTIONS.
POLYNOMIAL RINGS. RING OF POLYNOMIAL FUNCTIONS ON A PLANE ALGEBRAIC
CURVE. RING OF POWER SERIES AND FORMAL POWER SERIES. BOOLEAN RINGS.
DIRECT SUMS OF RINGS. RING OF CONTINUOUS FUNCTIONS. FACTORISATION;
UNIQUE FACTORISATION DOMAINS, EXAMPLES OF UFDS. § 4. HOMOMORPHISMS AND
IDEALS 24 HOMOMORPHISMS, IDEALS, QUOTIENT RINGS. THE HOMOMORPHISMS
THEOREM. THE RESTRIC- TION HOMOMORPHISM IN RINGS OF FUNCTIONS. PRINCIPAL
IDEAL DOMAINS; RELATIONS WITH UFDS. PRODUCT OF IDEALS. CHARACTERISTIC OF
A FIELD. EXTENSION IN WHICH A GIVEN POLY- NOMIAL HAS A ROOT.
ALGEBRAICALLY CLOSED FIELDS. FINITE FIELDS. REPRESENTING ELEMENTS OF A
GENERAL RING AS FUNCTIONS ON MAXIMAL AND PRIME IDEALS. INTEGERS AS
FUNCTIONS. ULTRAPRODUCTS AND NONSTANDARD ANALYSIS. COMMUTING
DIFFERENTIAL OPERATORS. §5. MODULES 33 DIRECT SUMS AND FREE MODULES.
TENSOR PRODUCTS. TENSOR, SYMMETRIC AND EXTERIOR POWERS OF A MODULE, THE
DUAL MODULE. EQUIVALENT IDEALS AND ISOMORPHISM OF MODULES. MODULES OF
DIFFERENTIAL FORMS AND VECTOR FIELDS. FAMILIES OF VECTOR SPACES AND
MODULES. § 6. ALGEBRAIC ASPECTS OF DIMENSION 41 RANK OF A MODULE.
MODULES OF FINITE TYPE. MODULES OF FINITE TYPE OVER A PRINCIPAL IDEAL
DOMAIN. NOETHERIAN MODULES AND RINGS. NOETHERIAN RINGS AND RINGS OF
FINITE TYPE. THE CASE OF GRADED RINGS. TRANSCENDENCE DEGREE OF AN
EXTENSION. FINITE EXTENSIONS. 2 CONTENTS § 7. THE ALGEBRAIC VIEW OF
INFINITESIMAL NOTIONS 50 FUNCTIONS MODULO SECOND ORDER INFINITESIMALS
AND THE TANGENT SPACE OF A MANIFOLD. SINGULAR POINTS. VECTOR FIELDS AND
FIRST ORDER DIFFERENTIAL OPERATORS. HIGHER ORDER INFINITESIMALS. JETS
AND DIFFERENTIAL OPERATORS. COMPLETIONS OF RINGS, P-ADIC NUMBERS. NORMED
FIELDS. VALUATIONS OF THE FIELDS OF RATIONAL NUMBERS AND RATIONAL
FUNCTIONS. THE P-ADIC NUMBER FIELDS IN NUMBER THEORY. § 8.
NONCOMMUTATIVE RINGS 61 BASIC DEFINITIONS. ALGEBRAS OVER RINGS. RING OF
ENDOMORPHISMS OF A MODULE. GROUP ALGEBRA. QUATERNIONS AND DIVISION
ALGEBRAS. TWISTOR FIBRATION. ENDOMORPHISMS OF N-DIMENSIONAL VECTOR SPACE
OVER A DIVISION ALGEBRA. TENSOR ALGEBRA AND THE NON- COMMUTING
POLYNOMIAL RING. EXTERIOR ALGEBRA; SUPERALGEBRAS; CLIFFORD ALGEBRA.
SIMPLE RINGS AND ALGEBRAS. LEFT AND RIGHT IDEALS OF THE ENDOMORPHISM
RING OF A VECTOR SPACE OVER A DIVISION ALGEBRA. § 9. MODULES OVER
NONCOMMUTATIVE RINGS 74 MODULES AND REPRESENTATIONS. REPRESENTATIONS OF
ALGEBRAS IN MATRIX FORM. SIMPLE MODULES, COMPOSITION SERIES, THE
JORDAN-HOLDER THEOREM. LENGTH OF A RING OR MODULE. ENDOMORPHISMS OF A
MODULE. SCHUR S LEMMA § 10. SEMISIMPLE MODULES AND RINGS 79
SEMISIMPLICITY. A GROUP ALGEBRA IS SEMISIMPLE. MODULES OVER A SEMISIMPLE
RING. SEMI- SIMPLE RINGS OF FINITE LENGTH; WEDDERBURN S THEOREM. SIMPLE
RINGS OF FINITE LENGTH AND THE FUNDAMENTAL THEOREM OF PROJECTIVE
GEOMETRY. FACTORS AND CONTINUOUS GEOMETRIES. SEMISIMPLE ALGEBRAS OF
FINITE RANK OVER AN ALGEBRAICALLY CLOSED FIELD. APPLICATIONS TO
REPRESENTATIONS OF FINITE GROUPS. §11. DIVISION ALGEBRAS OF FINITE RANK
90 DIVISION ALGEBRAS OF FINITE RANK OVER U OR OVER FINITE FIELDS. TSEN S
THEOREM AND QUASI-ALGEBRAICALLY CLOSED FIELDS. CENTRAL DIVISION ALGEBRAS
OF FINITE RANK OVER THE P-ADIC AND RATIONAL FIELDS. § 12. THE NOTION OF
A GROUP 96 TRANSFORMATION GROUPS, SYMMETRIES, AUTOMORPHISMS. SYMMETRIES
OF DYNAMICAL SYS- TEMS AND CONSERVATION LAWS. SYMMETRIES OF PHYSICAL
LAWS. GROUPS, THE REGULAR ACTION. SUBGROUPS, NORMAL SUBGROUPS, QUOTIENT
GROUPS. ORDER OF AN ELEMENT. THE IDEAL CLASS GROUP. GROUP OF EXTENSIONS
OF A MODULE. BRAUER GROUP. DIRECT PRODUCT OF TWO GROUPS. § 13. EXAMPLES
OF GROUPS: FINITE GROUPS 108 SYMMETRIC AND ALTERNATING GROUPS. SYMMETRY
GROUPS OF REGULAR POLYGONS AND REGULAR POLYHEDRONS. SYMMETRY GROUPS OF
LATTICES. CRYSTALLOGRAPHIC CLASSES. FINITE GROUPS GENERATED BY
REFLECTIONS. § 14. EXAMPLES OF GROUPS: INFINITE DISCRETE GROUPS 124
DISCRETE TRANSFORMATION GROUPS. CRYSTALLOGRAPHIC GROUPS. DISCRETE GROUPS
OF MOTION OF THE LOBACHEVSKY PLANE. THE MODULAR GROUP. FREE GROUPS.
SPECIFYING A GROUP BY GENERATORS AND RELATIONS. LOGICAL PROBLEMS. THE
FUNDAMENTAL GROUP. GROUP OF A KNOT. BRAID GROUP. § 15. EXAMPLES OF
GROUPS: LIE GROUPS AND ALGEBRAIC GROUPS 140 LIE GROUPS. TORUSES. THEIR
ROLE IN LIOUVILLE S THEOREM. A. COMPACT LIE GROUPS 143 THE CLASSICAL
COMPACT GROUPS AND SOME OF THE RELATIONS BETWEEN THEM. B. COMPLEX
ANALYTIC LIE GROUPS 147 THE CLASSICAL COMPLEX LIE GROUPS. SOME OTHER LIE
GROUPS. THE LORENTZ GROUP. C. ALGEBRAIC GROUPS 150 ALGEBRAIC GROUPS, THE
ADELE GROUP. TAMAGAWA NUMBER. CONTENTS 3 § 16. GENERAL RESULTS OF GROUP
THEORY 151 DIRECT PRODUCTS. THE WEDDERBURN-REMAK-SHMIDT THEOREM.
COMPOSITION SERIES, THE JORDAN-HOLDER THEOREM. SIMPLE GROUPS, SOLVABLE
GROUPS. SIMPLE COMPACT LIE GROUPS. SIMPLE COMPLEX LIE GROUPS. SIMPLE
FINITE GROUPS, CLASSIFICATION. § 17. GROUP REPRESENTATIONS 160 A.
REPRESENTATIONS OF FINITE GROUPS 163 REPRESENTATIONS. ORTHOGONALITY
RELATIONS. B. REPRESENTATIONS OF COMPACT LIE GROUPS 167 REPRESENTATIONS
OF COMPACT GROUPS. INTEGRATING OVER A GROUP. HELMHOLTZ-LIE THEORY.
CHARACTERS OF COMPACT ABELIAN GROUPS AND FOURIER SERIES. WEYL AND RICCI
TENSORS IN 4- DIMENSIONAL RIEMANNIAN GEOMETRY. REPRESENTATIONS OF SU(2)
AND SO(3). ZEEMAN EFFECT. C. REPRESENTATIONS OF THE CLASSICAL COMPLEX
LIE GROUPS 174 REPRESENTATIONS OF NONCOMPACT LIE GROUPS. COMPLETE
IRREDUCIBILITY OF REPRESENTATIONS OF FINITE-DIMENSIONAL CLASSICAL
COMPLEX LIE GROUPS. § 18. SOME APPLICATIONS OF GROUPS 177 A. GALOIS
THEORY 177 GALOIS THEORY. SOLVING EQUATIONS BY RADICALS. B. THE GALOIS
THEORY OF LINEAR DIFFERENTIAL EQUATIONS (PICARD- VESSIOT THEORY) 181 C.
CLASSIFICATION OF UNRAMIFIED COVERS 182 CLASSIFICATION OF UNRAMIFIED
COVERS AND THE FUNDAMENTAL GROUP D. INVARIANT THEORY 183 THE FIRST
FUNDAMENTAL THEOREM OF INVARIANT THEORY E. GROUP REPRESENTATIONS AND THE
CLASSIFICATION OF ELEMENTARY PARTICLES 185 § 19. LIE ALGEBRAS AND
NONASSOCIATIVE ALGEBRA 188 A. LIE ALGEBRAS 188 POISSON BRACKETS AS AN
EXAMPLE OF A LIE ALGEBRA. LIE RINGS AND LIE ALGEBRAS. B. LIE THEORY 192
LIE ALGEBRA OF A LIE GROUP. C. APPLICATIONS OF LIE ALGEBRAS 197 LIE
GROUPS AND RIGID BODY MOTION. D. OTHER NONASSOCIATIVE ALGEBRAS 199 THE
CAYLEY NUMBERS. ALMOST COMPLEX STRUCTURE ON 6-DIMENSIONAL SUBMANIFOLDS
OF 8-SPACE. NONASSOCIATIVE REAL DIVISION ALGEBRAS. §20. CATEGORIES 202
DIAGRAMS AND CATEGORIES. UNIVERSAL MAPPING PROBLEMS. FUNCTORS. FUNCTORS
ARISING IN TOPOLOGY: LOOP SPACES, SUSPENSION. GROUP OBJECTS IN
CATEGORIES. HOMOTOPY GROUPS. §21. HOMOLOGICAL ALGEBRA 213 A. TOPOLOGICAL
ORIGINS OF THE NOTIONS OF HOMOLOGICAL ALGEBRA .. . 213 COMPLEXES AND
THEIR HOMOLOGY. HOMOLOGY AND COHOMOLOGY OF POLYHEDRONS. FIXED POINT
THEOREM. DIFFERENTIAL FORMS AND DE RHAM COHOMOLOGY; DE RHAM S THEOREM.
LONG EXACT COHOMOLOGY SEQUENCE. B. COHOMOLOGY OF MODULES AND GROUPS 219
COHOMOLOGY OF MODULES. GROUP COHOMOLOGY. TOPOLOGICAL MEANING OF THE
COHO- MOLOGY OF DISCRETE GROUPS. C. SHEAF COHOMOLOGY 225 SHEAVES; SHEAF
COHOMOLOGY. FINITENESS THEOREMS. RIEMANN-ROCH THEOREM. 4 PREFACE §22.
X-THEORY 230 A. TOPOLOGICAL K-THEORY 230 VECTOR BUNDLES AND THE FUNCTOR
Y E C(X). PERIODICITY AND THE FUNCTORS K N (X). K X (X) AND THE
INFINITE-DIMENSIONAL LINEAR GROUP. THE SYMBOL OF AN ELLIPTIC
DIFFERENTIAL OPERATOR. THE INDEX THEOREM. B. ALGEBRAIC K-THEORY 234 THE
GROUP OF CLASSES OF PROJECTIVE MODULES. K O , K, AND K N OF A RING. K 2
OF A FIELD AND ITS RELATIONS WITH THE BRAUER GROUP. K-THEORY AND
ARITHMETIC. COMMENTS ON THE LITERATURE 239 REFERENCES 244 INDEX OF NAMES
249 SUBJECT INDEX 251
|
| any_adam_object | 1 |
| author | Šafarevič, Igorʹ R. 1923-2017 |
| author_GND | (DE-588)119280337 |
| author_facet | Šafarevič, Igorʹ R. 1923-2017 |
| author_role | aut |
| author_sort | Šafarevič, Igorʹ R. 1923-2017 |
| author_variant | i r š ir irš |
| building | Verbundindex |
| bvnumber | BV011068116 |
| classification_rvk | SK 200 |
| classification_tum | MAT 110f |
| ctrlnum | (OCoLC)247041771 (DE-599)BVBBV011068116 |
| discipline | Mathematik |
| edition | 2. printing |
| format | Book |
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| id | DE-604.BV011068116 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:03:28Z |
| institution | BVB |
| isbn | 3540612211 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007413581 |
| oclc_num | 247041771 |
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| owner | DE-20 DE-384 DE-703 DE-824 DE-91G DE-BY-TUM DE-83 DE-188 DE-634 |
| owner_facet | DE-20 DE-384 DE-703 DE-824 DE-91G DE-BY-TUM DE-83 DE-188 DE-634 |
| physical | 258 S. Ill., graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Springer |
| record_format | marc |
| spelling | Šafarevič, Igorʹ R. 1923-2017 Verfasser (DE-588)119280337 aut Basic notions of algebra I. R. Shafarevich 2. printing Berlin [u.a.] Springer 1997 258 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Auch als: Algebra ; 1 Algebra (DE-588)4001156-2 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 1. Auflage Algebra GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007413581&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Šafarevič, Igorʹ R. 1923-2017 Basic notions of algebra Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4001156-2 |
| title | Basic notions of algebra |
| title_auth | Basic notions of algebra |
| title_exact_search | Basic notions of algebra |
| title_full | Basic notions of algebra I. R. Shafarevich |
| title_fullStr | Basic notions of algebra I. R. Shafarevich |
| title_full_unstemmed | Basic notions of algebra I. R. Shafarevich |
| title_old | Algebra |
| title_short | Basic notions of algebra |
| title_sort | basic notions of algebra |
| topic | Algebra (DE-588)4001156-2 gnd |
| topic_facet | Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007413581&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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