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