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

...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
1
12 Mappings Cardinality
2
13 The Number Sequence
3
14 Finite and Countable Denumerable Sets
7
15 Partitions
10
GROUPS
12
22 Subgroups
19
23 Complexes Cosets
23
67 Galois Fields Finite Commutative Fields
129
68 Separable and Inseparable Extensions
133
69 Perfect and Imperfect Fields
137
610 Simplicity of Algebraic Extensions Theorem on the Primitive Element
139
611 Norms and Traces
140
CONTINUATION OF GROUP THEORY
144
72 Operator Isomorphisms and Operator Homomorphisms
146
73 The Two Laws of Isomorphism
147

24 Isomorphisms and Automorphisms
25
25 Homomorphisms Normal Subgroups and Factor Groups
28
RINGS AND FIELDS
32
32 Homomorphisms and Isomorphism
39
33 The Concept of a Field of Quotients
40
34 Polynomial Rings
43
35 Ideals Residue Class Rings
47
36 Divisibility Prime Ideals
51
37 Euclidean Rings and Principal Ideal Rings
53
38 Factorization
57
VECTOR SPACES AND TENSOR SPACES
61
42 Dimensional Invariance
64
43 The Dual Vector Space
66
44 Linear Equations in a Skew Field
68
45 Linear Transformations
69
46 Tensors
74
47 Antisymmetric Multilinear Forms and Determinants
76
48 Tensors
80
POLYNOMIALS
83
52 The Zeros of a Polynominal
85
53 Interpolation Formulae
86
54 Factorization
91
55 Irreducibility Criteria
94
56 Factorization in a Finite Number of Steps
97
57 Symmetric Functions
99
58 The Resultant of Two Polynominals
102
59 The Resultant as a Symmetric Function of the Roots
105
510 Partial Fraction Decomposition
107
THEORY OF FIELDS
110
62 Adjunction
112
63 Simple Field Extensions
113
64 Finite Field Extensions
118
65 Algebraic Field Extensions
120
66 Roots of Unity
125
74 Normal Series and Composition Series
148
75 Groups of Order pⁿ
152
76 Direct Products
153
77 Group Characters
156
78 Simplicity of the Alternating Group
160
79 Transitivity and Primitivity
162
THE GALOIS THEORY
165
82 The Fundamental Theorem of the Galois Theory
168
83 Conjugate Groups Conjugate Fields and Elements
170
84 Cyclotomic Fields
172
85 Cyclic Fields and Pure Equations
178
86 Solution of Equations by Radicals
181
87 The General Equation of Degree n
184
88 Equations of the Second Third and Fourth Degrees
187
89 Constructions with Ruler and Compass
193
810 Calculation of the Galois Group Equations with a Symmetric Group
197
811 Normal Bases
200
ORDERING AND WELL ORDERING OF SETS
205
92 The Axiom of Choice and Zorns Lemma
206
93 The WellOrdering Theorem
209
INFINITE FIELD EXTENSIONS
212
102 Simple Transcendental Extensions
217
103 Algebraic Dependence and Independence
220
104 The Degree of Transcendency
223
105 Differentiation of Algebraic Functions
225
REAL FIELDS
231
112 Definition of the Real Numbers
234
113 Zeros of Real Functions
242
114 The Field of Complex Numbers
246
115 Algebraic Theory of Real Fields
248
116 Existence Theorems for Formally Real Fields
253
117 Sums of Squares
256
Index
258
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