Book:Nathan Jacobson/Lectures in Abstract Algebra
Jump to navigation
Jump to search
Nathan Jacobson: Lectures in Abstract Algebra
Published $\text {1951 - 1964}$
Subject Matter
Volume $\text I$: Basic Concepts
- Preface
- Introduction: Concepts from Set Theory: The System of Natural Numbers
- 1. Operations on sets
- 2. Product sets, mappings
- 3. Equivalence relations
- 4. The natural numbers
- 5. The system of integers
- 6. The division process in $I$
- Chapter $\text I$: Semi-Groups and Groups
- 1. Definition and examples of semi-groups
- 2. Non-associative binary compositions
- 3. Generalized associative law. Powers
- 4. Commutativity
- 5. Identities and inverses
- 6. Definition and examples of groups
- 7. Subgroups
- 8. Isomorphism
- 9. Transformation groups
- 10. Realization of a group as a transformation group
- 11. Cyclic groups. Order of an element
- 12. Elementary properties of permutations
- 13. Coset decompositions of groups
- 14. Invariant subgroups and factor groups
- 15. Homomorphism of groups
- 16. The fundamental theorem of homomorphism for groups
- 17. Endomorphisms, automorphisms, center of a group
- 18. Conjugate classes
- Chapter $\text {II}$: Rings, Integral Domains and Fields
- 1. Definition and examples
- 2. Types of rings
- 3. Quasi-regularity. The circle composition
- 4. Matrix rings
- 5. Quaternions
- 6. Subrings generated by a set of elements. Center
- 7. Ideals, difference rings
- 8. Ideals and difference rings for the ring of integers
- 9. Homomorphism of rings
- 10. Anti-isomorphism
- 11. Structure of the additive group of a ring. The characteristic of a ring
- 12. Algebra of subgroups of the additive group of a ring. One-sided ideals
- 13. The ring of endomorphisms of a commutative group
- 14. The multiplications of a ring
- Chapter $\text {III}$: Extensions of a Ring and Fields
- 1. Imbedding of a ring in a ring with an identity
- 2. Fields of fractions on a commutative integral domain
- 3. Uniqueness of the field of fractions
- 4. Polynomial rings
- 5. Structure of polynomial rings
- 6. Properties of the ring $\mathfrak{A} \left[{ x }\right]$
- 7. Simple extensions of a field
- 8. Structure of any field
- 9. The number of roots of a polynomial in a field
- 10. Polynomials in several elements
- 11. Symmetric polynomials
- 12. Rings of functions
- Chapter $\text {IV}$: Elementary Factorization Theory
- 1. Factors, associates, irreducible elements
- 2. Gaussian semi-groups
- 3. Greatest common divisors
- 4. Principal ideal domains
- 5. Euclidean domains
- 6. Polynomial extensions of Gaussian domains
- Chapter $\text V$: Groups with Operators
- 1. Definition and examples of groups with operators
- 2. M-subgroups, M-factor groups and M-homomorphisms
- 3. The fundamental theorem of homomorphisms for M-groups
- 4. The correspondence between M-subgroups determined by a homomorphism
- 5. The isomorphism theorems for M-groups
- 6. Schreier's theorem
- 7. Simple groups and the Jordan-Hölder theorem
- 8. The chain conditions
- 9. Direct products
- 10. Direct products of subgroups
- 11. Projections
- 12. Decomposition into indecomposable groups
- 13. The Krull-Schmidt theorem
- 14. Infinite direct products
- Chapter $\text {VI}$: Modules and Ideals
- 1. Definitions
- 2. Fundamental concepts
- 3. Generators. Unitary modules
- 4. The chain conditions
- 5. The Hilbert basis theorem
- 6. Noetherian rings. Prime and primary ideals
- 7. Representation of an ideal as intersection of primary ideals
- 8. Uniqueness theorems
- 9. Integral dependence
- 10. Integers of quadratic fields
- Chapter $\text {VII}$: Lattices
- 1. Partially ordered sets
- 2. Lattices
- 3. Modular lattices
- 4. Schreier's theorem. The chain conditions
- 5. Decomposition theory for lattices with ascending chain condition
- 6. Independence
- 7. Complemented modular lattices
- 8. Boolean algebras
- Index
Volume $\text {II}$: Linear Algebra
- Preface
- Chapter $\text I$: Finite Dimensional Vector Spaces
- 1. Abstract vector spaces
- 2. Right vector spaces
- 3. $\mathfrak o$-modules
- 4. Linear dependence
- 5. Invariance of dimensionality
- 6. Bases and matrices
- 7. Applications to matrix theory
- 8. Rank of a set of vectors
- 9. Factor spaces
- 10. Algebra of subspaces
- 11. Independent subspaces, direct sums
- Chapter $\text {II}$: Linear Transformations
- 1. Definition and examples
- 2. Composition of linear transformations
- 3. The matrix of a linear transformation
- 4. Composition of matrices
- 5. Change of basis. Equivalence and similarity of matrices
- 6. Rank space and null space of a linear transformation
- 7. Systems of linear equations
- 8. Linear transformations in right vector spaces
- 9. Linear functions
- 10. Duality between a finite dimensional linear space and its conjugate space
- 11. Transpose of a linear transformation
- 12. Matrices of the transpose
- 13. Projections
- Chapter $\text {III}$: The Theory of a Single Linear Transformation
- 1. The minimum polynomial of a linear transformation
- 2. Cyclic subspaces
- 3. Existence of a vector whose order is the minimum polynomial
- 4. Cyclic linear transformations
- 5. The $\Phi \left[{\lambda}\right]$-module determined by a linear transformation
- 6. Finitely generated $\mathfrak o$-modules, $\mathfrak o$, a principal ideal domain
- 7. Normalization of the generators of $\mathfrak F$ and of $\mathfrak R$
- 8. Equivalence of matrices with elements in a principal ideal domain
- 9. Structure of finitely generated $\mathfrak o$-modules
- 10. Invariance theorems
- 11. Decomposition of a vector space relative to a linear transformation
- 12. The characteristic and minimum polynomials
- 13. Direct proof of Theorem 13
- 14. Formal properties of the trace and the characteristic polynomial
- 15. The ring of $\mathfrak o$-endomorphisms of a cyclic $\mathfrak o$-module
- 16. Determination of the ring of $\mathfrak o$-endomorphisms of a finitely generated $\mathfrak o$-module, $\mathfrak o$ principal
- 17. The linear transformations which commute with a given linear transformation
- 18. The center of the ring $\mathfrak B$
- Chapter $\text {IV}$: Sets of Linear Transformations
- 1. Invariant subspaces
- 2. Induced linear transformations
- 3. Composition series
- 4. Decomposability
- 5. Complete reducibility
- 6. Relation to the theory of operator groups and the theory of modules
- 7. Reducibility, decomposability, complete reducibility for a single linear transformation
- 8. The primary components of a space relative to a linear transformation
- 9. Sets of commutative linear transformations
- Chapter $\text V$: Bilinear Forms
- 1. Bilinear forms
- 2. Matrices of a bilinear form
- 3. Non-degenerate forms
- 4. Transpose of a linear transformation relative to a pair of bilinear forms
- 5. Another relation between linear transformations and bilinear forms
- 6. Scalar products
- 7. Hermitian scalar products
- 8. Matrices of hermitian scalar products
- 9. Symmetric and hermitian scalar products over special division rings
- 10. Alternate scalar products
- 11. Witt's theorem
- 12. Non-alternate skew-symmetric forms
- Chapter $\text {VI}$: Euclidean and Unitary Spaces
- 1. Cartesian bases
- 2. Linear transformations and scalar products
- 3. Orthogonal complete reducibility
- 4. Symmetric, skew and orthogonal linear transformations
- 5. Canonical matrices for symmetric and skew linear transformations
- 6. Commutative symmetric and skew linear transformations
- 7. Normal and orthogonal linear transformations
- 8. Semi-definite transformations
- 9. Polar factorization of an arbitrary linear transformation
- 10. Unitary geometry
- 11. Analytic functions of linear transformations
- Chapter $\text {VII}$: Products of Vector Spaces
- 1. Product groups of vector spaces
- 2. Direct products of linear transformations
- 3. Two-sided vector spaces
- 4. The Kronecker product
- 5. Kronecker products of linear transformations and of matrices
- 6. Tensor spaces
- 7. Symmetry classes of tensors
- 8. Extension of the field of a vector space
- 9. A theorem on similarity of sets of matrices
- 10. Alternative definition of an algebra. Kronecker product of algebras
- Chapter $\text {VIII}$: The Ring of Linear Transformations
- 1. Simplicity of $\mathfrak L$
- 2. Operator methods
- 3. The left ideals of $\mathfrak L$
- 4. Right ideals
- 5. Isomorphisms of rings of linear transformations
- Chapter $\text {IX}$: Infinite Dimensional Vector Spaces
- 1. Existence of a basis
- 2. Invariance of dimensionality
- 3. Subspaces
- 4. Linear transformations and matrices
- 5. Dimensionality of the conjugate space
- 6. Finite topology for linear transformations
- 7. Total subspaces of $\mathfrak R^*$
- 8. Dual subspaces. Kronecker products
- 9. Two-sided ideals in the ring of linear transformations
- 10. Dense rings of linear transformations
- 11. Isomorphism theorems
- 12. Anti-automorphisms and scalar products
- 13. Schur's lemma. A general density theorem
- 14. Irreducible algebras of linear transformations
- Index
Volume $\text {III}$: The Theory of Fields and Galois Theory
![]() | A table of contents is missing for this source work. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding the table of contents. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ContentsWanted}} from the code. |
Source work progress
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Chapter $\text{I}$: Semi-Groups and Groups: $1$: Definition and examples of semigroups