Algebra

Start reading → Algebraic structures

1. Front matter
   1. Introduction
   2. Introduction
   3. To the reader
   4. To the reader
   5. To the reader
2. IAlgebraic structures
   1. Algebraic structures
   2. § 1. Laws of composition; associativity; commutativity31
   3. § 2. Identity element; cancellable elements; invertible elements26
   4. § 3. Actions14
   5. § 4. Groups and groups with operators63
   6. § 5. Groups operating on a set32
   7. § 6. Extensions, solvable groups, nilpotent groups58
   8. § 7. Free monoids, free groups17
   9. § 8. Rings32
   10. § 9. Fields17
   11. § 10. Inverse and direct limits5
   12. Historical Note
3. IILinear algebra
   1. Linear algebra
   2. § 1. Modules88
   3. § 2. Modules of linear mappings. Duality42
   4. § 3. Tensor products30
   5. § 4. Relations between tensor products and homomorphism modules15
   6. § 5. Extension of the ring of scalars13
   7. § 6. Inverse and direct limits of modules17
   8. § 7. Vector spaces87
   9. § 8. Restriction of the field ofscalars in vector spaces17
   10. § 9. Affine spaces and projective spaces12
   11. § 10. Matrices35
   12. § 11. Graded modules and rings28
   13. App. 0. Pseudomodules
4. IIITensor algebras, exterior algebras, symmetric algebras
   1. Tensor algebras, exterior algebras, symmetric algebras
   2. § 1. Algebras5
   3. § 2. Examples of algebras17
   4. § 3. Graded algebras4
   5. § 4. Tensor products of algebras43
   6. § 5. Tensor algebra. Tensors15
   7. § 6. Symmetric algebras17
   8. § 7. Exterior algebras31
   9. § 8. Determinants36
   10. § 9. Norms and traces13
   11. § 10. Derivations41
   12. § 11. Cogebras, products of multilinear forms, inner products and duality40
   13. App. 0. Alternative algebras. Octonions11
   14. Historical Note
5. IVPolynomials and rational fractions
   1. Polynomials and rational fractions
   2. § 1. Polynomials30
   3. § 2. Zeros of polynomials20
   4. § 3. Rational fractions6
   5. § 4. Formal power series23
   6. § 5. Symmetric tensors and polynomial mappings41
   7. § 6. Symmetric functions43
6. VCommutative fields
   1. Commutative fields
   2. § 1. Prime fields. Characteristic23
   3. § 2. Extensions19
   4. § 3. Algebraic extensions18
   5. § 4. Algebraically closed extensions20
   6. § 5. p-radical extensions8
   7. § 6. Etale algebras22
   8. § 7. Separable algebraic extensions39
   9. § 8. Norms and traces9
   10. § 9. Conjugate elements and quasi-Galois extensions19
   11. § 10. Galois extensions49
   12. § 11. Abelian extensions45
   13. § 12. Finite fields11
   14. § 13. p-radical extensions of height _ 121
   15. § 14. Transcendental extensions51
   16. § 15. Separable extensions21
   17. § 16. Differential criteria of separability32
   18. § 17. Regular extensions20
   19. Historical Note
7. VIOrdered groups and fields
   1. Ordered groups and fields
   2. § 1. Ordered groups. Divisibility53
   3. § 2. Ordered fields33
8. VIIModules over principal ideal domains
   1. Modules over principal ideal domains
   2. § 1. Principal ideal domains15
   3. § 2. Torsion modules over a principal ideal domain22
   4. § 3. Free modules over a principal ideal domain6
   5. § 4. Finitely generated modules over a principal ideal domain40
   6. § 5. Endomorphisms of vector spaces66
   7. Historical Note
9. VIIISemisimple Modules and Rings
   1. Semisimple Modules and Rings
   2. § 1. Artinian Modules and Noetherian Modules36
   3. § 2. The Structure of Modules of Finite Length37
   4. § 3. Simple Modules23
   5. § 4. Semisimple Modules43
   6. § 5. Commutation37
   7. § 6. Morita Equivalence of Modules and Algebras40
   8. § 7. Simple Rings24
   9. § 8. Semisimple Rings30
   10. § 9. Radical49
   11. § 10. Modules over an Artinian Ring12
   12. § 11. Grothendieck Groups38
   13. § 12. Tensor Products of Semisimple Modules26
   14. § 13. Absolutely Semisimple Algebras33
   15. § 14. Central Simple Algebras36
   16. § 15. Brauer Groups12
   17. § 16. Other Descriptions of the Brauer Group63
   18. § 17. Reduced Norms and Traces31
   19. § 18. Simple Algebras over a Finite Field10
   20. § 19. Quaternion Algebras16
   21. § 20. Linear Representations of Algebras31
   22. § 21. Linear Representations of Finite Groups36
   23. App. 1. Algebras without Unit Element20
   24. App. 2. Determinants over a Noncommutative Field18
   25. App. 3. Hilbert’s Nullstellensatz3
   26. App. 4. Trace of an Endomorphism of Finite Rank5
   27. Historical Note
10. Indexes
    1. Index of notation
    2. Index of notation
    3. Notation index
    4. Index of terminology
    5. Index of terminology
    6. Terminology index