Higher algebra,

Title Page......Page 1
Copyright Page......Page 4
Contents......Page 5
Volume I Linear Equations......Page 3
Introduction. The Basic Problem of Algebra......Page 9
1. Definition of Rings, Fields, Integral Domains......Page 13
2. Subdomains. Congruence Relations. Isomorphism......Page 20
3. The Quotient Field of an Integral Domain......Page 33
4. The Integral Domain of Integral Rational Functions of n Indeterminates over I and the Field of Rational Functions of n Indeterminates over K......Page 38
5. Detailed Formulation of the Basic Problem of Algebra......Page 53
6. Definition of Groups......Page 57
7. Subgroups. Congruence Relations. Isomorphism......Page 63
8. Partition of a Group Relative to a Subgroup......Page 66
9. Normal Divisors. Conjugate Subsets of a Group. Factor Groups......Page 69
10. Linear Forms. Vectors. Matrices......Page 79
11. Nonhomogeneous and Homogeneous Systems of Linear Equations......Page 89
12. The Toeplitz Process......Page 94
13. Solvability and Solutions of Systems of Linear Equations......Page 102
14. The Case m=n......Page 110
15. Importance of Linear Algebra without Determinants......Page 114
16. Permutation Groups......Page 117
17. Determinants......Page 127
18. Minors and Cofactors. The Laplace Expansion Theorem......Page 131
19. Further Theorems on Determinants......Page 140
20. Application of the Theory of Determinants to Systems of Linear Equations in the Case m=n......Page 144
21. The Rank of a Matrix......Page 149
22. Application of the Theory of Determinants to Systems of Linear Equations in the General Case......Page 157
Conclusion. Dependence on the Ground Field......Page 163
Volume II Equations of Higher Degree......Page 165
Introduction. Methodical Preliminary Observations and Survey......Page 167
1. The Fundamental Theorem of the Unique Decomposability into Prime Elements in K [x] and r......Page 172
2. Residue Class Rings in K [x] and r......Page 189
3. Cyclic Groups......Page 196
4. Prime Integral Domains. Prime Fields. Characteristic......Page 200
5. Roots and Linear Factors......Page 206
6. Multiple Roots. Derivative......Page 211
7. General Theory of Extensions (1). Basic Concepts and Facts......Page 218
8. Stem Fields......Page 230
9. General Theory of Extensions (2). Simple and Finite Algebraic Extensions......Page 235
10. Root Fields......Page 242
11. The So-called Fundamental Theorem of Algebra......Page 247
12. Simplicity and Separability of the Root Fields of Separable Polynomials, More Generally, of Finite Algebraic Extensions with Separable Primitive System of Elements......Page 250
13. Normality of Root Fields and of Their Primitive Elements. Galois Resolvents......Page 255
14. The Automorphism Group of an Extension Domain......Page 264
15. The Galois Group of a Separable Normal Extension of Finite Degree......Page 267
16. The Galois Group of a Separable Polynomial......Page 270
17. The Fundamental Theorem of Galois Theory......Page 273
18. Dependence on the Ground Field......Page 288
19. Definition of Solvability by Radicals......Page 301
20. Cyclotomic Fields. Finite Fields......Page 303
21. Pure and Cyclic Extensions of Prime Degree......Page 312
22. Criterion for the Solvability by Radicals......Page 318
23. Existence of Algebraic Equations not Solvable by Radicals......Page 323
Index......Page 333