Algebra: Based in Part on Lectures by E. Artin and E. Noether. ...

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Springer Science & Business Media, Oct 21, 2003 - Mathematics - 265 pages
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...This beautiful and eloquent text served to transform the graduate teaching of algebra, not only in Germany, but elsewhere in Europe and the United States. It formulated clearly and succinctly the conceptual and structural insights which Noether had expressed so forcefully. This was combined with the elegance and understanding with which Artin had lectured...Its simple but austere style set the pattern for mathematical texts in other subjects, from Banach spaces to topological group theory...It is, in my view, the most influential text in algebra of the twentieth century.

- Saunders MacLane, Notices of the AMS

How exciting it must have been to hear Emil Artin and Emmy Noether lecture on algebra in the 1920's, when the axiomatic approach to the subject was amazing and new! Van der Waerden was there, and produced from his notes the classic textbook of the field. To Artin's clarity and Noether's originality he added his extraordinary gift for synthesis. At one time every would-be algebraist had to study this text. Even today, all who work in Algebra owe a tremendous debt to it; they learned from it by second or third hand, if not directly. It is still a first-rate (some would say, the best) source for the great range of material it contains.

- David Eisenbud, Mathematical Sciences Research Institute

Van der Waerden's book Moderne Algebra, first published in 1930, set the standard for the unified approach to algebraic structures in the twentieth century. It is a classic, still worth reading today.

- Robin Hartshorne, University of California, Berkeley

 

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Contents

NUMBERS AND SETS
xiii
12 Mappings Cardinality
xiv
13 The Number Sequence
1
14 Finite and Countable Denumerable Sets
5
15 Partitions
8
GROUPS
10
22 Subgroups
17
23 Complexes Cosets
21
67 Galois Fields Finite Commutative Fields
127
68 Separable and Inseparable Extensions
131
69 Perfect and Imperfect Fields
135
610 Simplicity of Algebraic Extensions Theorem on the Primitive Element
137
611 Norms and Traces
138
CONTINUATION OF GROUP THEORY
142
72 Operator Isomorphisms and Operator Homomorphisms
144
73 The Two Laws of Isomorphism
145

24 Isomorphisms and Automorphisms
23
25 Homomorphisms Normal Subgroups and Factor Groups
26
RINGS AND FIELDS
30
32 Homomorphisms and Isomorphism
37
33 The Concept of a Field of Quotients
38
34 Polynomial Rings
41
35 Ideals Residue Class Rings
45
36 Divisibility Prime Ideals
49
37 Euclidean Rings and Principal Ideal Rings
51
38 Factorization
55
VECTOR SPACES AND TENSOR SPACES
59
42 Dimensional Invariance
62
43 The Dual Vector Space
64
44 Linear Equations in a Skew Field
66
45 Linear Transformations
67
46 Tensors
72
47 Antisymmetric Multilinear Forms and Determinants
74
48 Tensors
78
POLYNOMIALS
81
52 The Zeros of a Polynominal
83
53 Interpolation Formulae
84
54 Factorization
89
55 Irreducibility Criteria
92
56 Factorization in a Finite Number of Steps
95
57 Symmetric Functions
97
58 The Resultant of Two Polynominals
100
59 The Resultant as a Symmetric Function of the Roots
103
510 Partial Fraction Decomposition
105
THEORY OF FIELDS
108
62 Adjunction
110
63 Simple Field Extensions
111
64 Finite Field Extensions
116
65 Algebraic Field Extensions
118
66 Roots of Unity
123
74 Normal Series and Composition Series
146
75 Groups of Order pⁿ
150
76 Direct Products
151
77 Group Characters
154
78 Simplicity of the Alternating Group
158
79 Transitivity and Primitivity
160
THE GALOIS THEORY
163
82 The Fundamental Theorem of the Galois Theory
166
83 Conjugate Groups Conjugate Fields and Elements
168
84 Cyclotomic Fields
170
85 Cyclic Fields and Pure Equations
176
86 Solution of Equations by Radicals
179
87 The General Equation of Degree n
182
88 Equations of the Second Third and Fourth Degrees
185
89 Constructions with Ruler and Compass
191
810 Calculation of the Galois Group Equations with a Symmetric Group
195
811 Normal Bases
198
ORDERING AND WELL ORDERING OF SETS
203
92 The Axiom of Choice and Zorns Lemma
204
93 The WellOrdering Theorem
207
INFINITE FIELD EXTENSIONS
210
102 Simple Transcendental Extensions
215
103 Algebraic Dependence and Independence
218
104 The Degree of Transcendency
221
105 Differentiation of Algebraic Functions
223
REAL FIELDS
229
112 Definition of the Real Numbers
232
113 Zeros of Real Functions
240
114 The Field of Complex Numbers
244
115 Algebraic Theory of Real Fields
246
116 Existence Theorems for Formally Real Fields
251
117 Sums of Squares
254
Index
256
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