Basic algebra: 1
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Mineola, N.Y.
Dover Publications
2009
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 499 S. graph. Darst. |
| ISBN: | 9780486471891 0486471896 |
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MARC
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| 245 | 1 | 0 | |a Basic algebra |n 1 |c Nathan Jacobson |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Mineola, N.Y. |b Dover Publications |c 2009 | |
| 300 | |a XVIII, 499 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804141595846508544 |
|---|---|
| adam_text | Contents
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
natura]
numbers
15
0.5
The number system
Z
of integers
Î
9
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
Vill
Contents
1.4
Generalized associativity.
Commuta
ti vi ty
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
2
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.Ld.
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
Contents ix
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
»-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
ρ
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
Tarsia 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
36
í
6.5
Witt s cancellation theorem
367
6.6
The theorem of
Cartan-Dieudonné
371
6.7
Structure of the general linear group GLJF)
375
6.8
Structure of orthogonal groups
382
6.9
Sympłectic
geometry. The symplectic group
391
6.10
Orders of orthogonal and symplectic groups over a finite field
398
6.11
Postscript on hermman forms and unitary geometry
401
7
ALGEBRAS OVER A FIELD
405
7.1
Definition and examples of associative algebras
406
7.2
Exterior algebras. Application to determinants
411
Contents
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
Möbius
function of a partially ordered set
480
Appendix
489
Index
493
|
| any_adam_object | 1 |
| author | Jacobson, Nathan 1910-1999 |
| author_GND | (DE-588)119081873 |
| author_facet | Jacobson, Nathan 1910-1999 |
| author_role | aut |
| author_sort | Jacobson, Nathan 1910-1999 |
| author_variant | n j nj |
| building | Verbundindex |
| bvnumber | BV025742639 |
| classification_rvk | SK 200 |
| ctrlnum | (OCoLC)902794930 (DE-599)BVBBV025742639 |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV025742639 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:18:23Z |
| institution | BVB |
| isbn | 9780486471891 0486471896 |
| language | English |
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| owner_facet | DE-11 DE-355 DE-BY-UBR DE-898 DE-BY-UBR DE-29T |
| physical | XVIII, 499 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Dover Publications |
| record_format | marc |
| spelling | Jacobson, Nathan 1910-1999 Verfasser (DE-588)119081873 aut Basic algebra 1 Nathan Jacobson 2. ed. Mineola, N.Y. Dover Publications 2009 XVIII, 499 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebra (DE-588)4001156-2 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 (DE-604)BV024973775 1 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019346971&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jacobson, Nathan 1910-1999 Basic algebra Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4001156-2 |
| title | Basic algebra |
| title_auth | Basic algebra |
| title_exact_search | Basic algebra |
| title_full | Basic algebra 1 Nathan Jacobson |
| title_fullStr | Basic algebra 1 Nathan Jacobson |
| title_full_unstemmed | Basic algebra 1 Nathan Jacobson |
| title_short | Basic algebra |
| title_sort | basic 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=019346971&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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