# Book:Nathan Jacobson/Lectures in Abstract Algebra

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## Nathan Jacobson:

## 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-HÃ¶lder 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

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## 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