The Elements of Coordinate Geometry by Loney

[micromass]

• Author: H.M. Schey
• Title: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
• Amazon Link: https://www.amazon.com/dp/0393925161/?tag=pfamazon01-20
• Prerequisities: Calculus 1,2,3

Table of Contents:

[LIST]
[*] Preface
[*] Introduction, Vector Functions, and Electrostatics
[LIST]
[*] Introduction
[*] Vector Functions
[*] Electrostatics
[*] Problems
[/LIST]
[*] Surface Integrals and the Divergence
[LIST]
[*] Gauss' Law
[*] The Unit Normal Vector
[*] Definition of Surface Integrals
[*] Evaluating Surface Integrals
[*] Flux
[*] Using Gauss' Law to Find the Field
[*] The Divergence
[*] The Divergence in Cylindrical and Spherical Coordinates
[*] The Del Notation
[*] The Divergence Theorem
[*] Two Simple Applications of the Divergence Theorem
[*] Problems
[/LIST]
[*] Line Integrals and the Curl
[LIST]
[*] Work and Line Integrals
[*] Line Integrals Involving Vector Functions
[*] Path Independence
[*] The Curl
[*] The Curl in Cylindrical and Spherical Coordinates
[*] The Meaning of the Curl
[*] Differential Form of the Circulation Law
[*] Stokes' Theorem
[*] An Application of Stokes' Theorem
[*] Stokes' Theorem and Simply Connected Regions
[*] Path Independence and the Curl
[*] Problems
[/LIST]
[*] The Gradient
[LIST]
[*] Line Integrals and the Gradient
[*] Finding the Electrostatic Field
[*] Using Laplace's Equation
[*] Directional Derivatives and the Gradient
[*] Geometric Significance of the Gradient
[*] The Gradient in Cylindrical and Spherical Coordinates
[*] Problems
[/LIST]
[*] Solutions to Problems
[*] Bibliography
[*] Index
[/LIST]

 

[micromass]

• Author: Sterling Berberian
• Title: Measure and Integration
• Amazon Link: https://www.amazon.com/dp/0821853287/?tag=pfamazon01-20
• Prerequisities:

Table of Contents:

[LIST]
[*] Preface
[*] Index of Symbols
[*] Measures
[LIST]
[*] Set Theoretic Notations and Terminology
[*] Rings and [itex]\sigma[/itex]-Rings
[*] The Lemma on Monotone Classes
[*] Set Functions, Measures
[*] Some Properties of Measures
[*] Outer Measures
[*] Extensions of Measures
[*] Lebesgue Measure
[*] Measurable Covers
[*] Completion of a Measure
[*] The LUB of an Increasingly Directed Family of Measures
[/LIST]
[*] Measurable Functions
[LIST]
[*] Measurable Spaces
[*] Measurable Functions
[*] Combinations of Measurable Functions
[*] Limits of Measurable Functions
[*] Localization of Measurability
[*] Simple Functions
[/LIST]
[*] Sequences of Measurable Functions
[LIST]
[*] Measure Spaces
[*] The "Almost Everywhere" Concept
[*] Almost Everywhere Convergence
[*] Convergence in Measure
[*] Almost Uniform Convergence, Egoroff's Theorem
[/LIST]
[*] Integrable Functions
[LIST]
[*] Integrable Simple Functions
[*] Heuristics
[*] Nonnegative Integrable Functions
[*] Integrable Functions
[*] Indefinite Integrals
[*] The Monotone Convergence Theorem
[*] Mean Convergence
[/LIST]
[*] Convergence Theorems
[LIST]
[*] Dominated Convergence in Measure
[*] Dominated Convergence Almost Everywhere
[*] The [itex]\mathcal{L}^1[/itex] Completeness Theorem
[*] Fatou's Lemma
[*] The Space [itex]\mathcal{L}^2[/itex], Riesz-Fisher Theorem
[/LIST]
[*] Product Measures
[LIST]
[*] Rectangles
[*] Cartesian Product of Two Measurable Spaces
[*] Sections
[*] Preliminaries
[*] The Product of Two Finite Measure Spaces
[*] The Product of Any Two Measure Spaces
[*] Product of Two [itex]\sigma[/itex]-Finite Measure Spaces; Iterated Integrals
[*] Fubini's Theorem
[*] Complements
[/LIST]
[*] Finite Signed Measures
[LIST]
[*] Absolute Continuity
[*] Finite Signed Measures
[*] Contractions of a Finite Signed Measure
[*] Purely Positive and Purely Negative Sets
[*] Comparison of Finite Measures
[*] A Preliminary Radon-Nikodym Theorem
[*] Jordan-Hahn Decomposition of a Finite Signed Measure
[*] Domination of a Finite Signed Measures
[*] The Radon-Nikodym Theorem for a Finite Measure Space
[*] The Radon-Nikodym Theorem for a [itex]\sigma[/itex]-Finite Measure Space
[*] Riesz Representation Theorem
[/LIST]
[*] Integration over Locally Compact Spaces
[LIST]
[*] Continuous Functions with Compact Support
[*] [itex]G_\delta[/itex]'s and [itex]F_\sigma[/itex]'s
[*] Baire Sets
[*] Borel SEts
[*] Preliminaries on Rings
[*] Regularity
[*] Regularity of Baire Measures
[*] Regularity (Continuous)
[*] Regular Borel Measures
[*] Contents
[*] Regular Contents
[*] The Regular Borel Extension of a Baire Measure
[*] Integration of Continuous Functions with Compact Support
[*] Approximation of Baire Functions
[*] Approximation of Borel Functions
[*] The Riesz-Markoff Representation Theorem
[*] Completion Regularity
[/LIST]
[*] Integration over Locally Compact Groups
[LIST]
[*] Topological Groups
[*] Translates, Haar Integrals
[*] Translation Rations
[*] Existence of a Haar Integral
[*] A Topological Lemma
[*] Uniqueness of the Haar Integral
[*] The Modular Function
[*] Haar Measure
[*] Translates of Integrable Functions
[*] Adjoints of Continuous Functions with Compact Support
[*] Convolution of Continuous Functons with Compact Support
[*] Adjoint of Integrable Functon
[*] The Operation f Triangle g
[*] Convolution of Integrable Baire Functions
[*] Associavity of Convolution
[*] The Group algebra
[*] Convolution of Integrable Simple Baire Functions
[*] The domain f*g
[*] Convolution of Integrable Borel Functions
[*] Complements on Haar Measure
[/LIST]
[*] References and Notes
[*] Bibliography
[*] Index
[/LIST]

 

[micromass]

• Author: Martin Aigner, Gunter Ziegler
• Title: Proofs from THE BOOK
• Amazon link https://www.amazon.com/dp/3642008550/?tag=pfamazon01-20
• Prerequisities: Being acquainted with proofs and rigorous mathematics.
• Level: Undergrad and above

Table of Contents:

[LIST]
[*] Number Theory
[LIST]
[*] Six proofs of the infinity of primes
[*] Bertrand’s postulate
[*] Binomial coefficients are (almost) never powers
[*] Representing numbers as sums of two squares
[*] The law of quadratic reciprocity
[*] Every finite division ring is a field
[*] Some irrational numbers
[*] Three times \pi^2/6
[/LIST]
[*] Geometry
[LIST]
[*] Hilbert’s third problem: decomposing polyhedra
[*] Lines in the plane and decompositions of graphs
[*] The slope problem
[*] Three applications of Euler’s formula
[*] Cauchy’s rigidity theorem
[*] Touching simplices
[*] Every large point set has an obtuse angle
[*] Borsuk’s conjecture
[/LIST]
[*] Analysis
[LIST]
[*] Sets, functions, and the continuum hypothesis
[*] In praise of inequalities
[*] The fundamental theorem of algebra
[*] One square and an odd number of triangles
[*] A theorem of Pólya on polynomials
[*] On a lemma of Littlewood and Offord
[*] Cotangent and the Herglotz trick
[*] Buffon’s needle problem
[/LIST]
[*] Combinatorics
[LIST]
[*] Pigeon-hole and double counting
[*] Tiling rectangles
[*] Three famous theorems on finite sets
[*] Shuffling cards
[*] Lattice paths and determinants
[*] Cayley’s formula for the number of trees
[*] Identities versus bijections
[*] Completing Latin squares
[/LIST]
[*] Graph Theory
[LIST]
[*] The Dinitz problem
[*] Five-coloring plane graphs
[*] How to guard a museum
[*] Turán’s graph theorem
[*] Communicating without errors
[*] The chromatic number of Kneser graphs
[*] Of friends and politicians
[*] Probability makes counting (sometimes) easy
[/LIST]
[*] About the Illustrations
[*] Index
[/LIST]

 

[micromass]

• Author: Karel Hrbacek, Thomas Jech
• Title: Introduction to Set Theory
• Amazon Link: https://www.amazon.com/dp/0824779150/?tag=pfamazon01-20
• Prerequisities: Proofs
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] Sets 
[LIST]
[*] Introduction to Sets
[*] Properties
[*] The Axioms
[*] Elementary Operations on gets
[/LIST]
[*] Relations, Functions, and Orderings
[LIST]
[*] Ordered Pairs
[*] Relations
[*] Functions
[*] Equivalences and Partitions
[*] Orderings
[/LIST]
[*] Natural Numbers
[LIST]
[*] Introduction to Natural Numbers
[*] Properties of Natural Numbers
[*] The Recursion Theorem
[*] Arithmetic of Natural Numbers
[*] Operations and Structures
[/LIST]
[*] Finite, Countable, and Uncountable Sets 
[LIST]
[*] Cardinality of gets
[*] Finite Sets
[*] Countable gets
[*] Linear Orderings
[*] Complete Linear Orderings
[*] Uncountable gets
[/LIST]
[*] Cardinal Numbers
[LIST]
[*] Cardinal Arithmetic
[*] The Cardinality of the Continuum
[/LIST]
[*] Ordinal Numbers
[LIST]
[*] Well-Ordered Sets
[*] Ordinal Numbers
[*] The Axiom of Replacement
[*] Transfinite Induction and Recursion
[*] Ordinal Arithmetic
[*] The Normal Form
[/LIST]
[*] Alephs
[LIST]
[*] Initial Ordinals
[*] Addition and Multiplication of Alephs
[/LIST]
[*] The Axiom of Choice
[LIST]
[*] The Axiom of Choice and its Equivalents
[*] The Use of the Axiom of Choice in Mathematics
[/LIST]
[*] Arithmetic of Cardinal Numbers
[LIST]
[*] Infinite Sums and Products of Cardinal Numbers
[*] Regular and Singular Cardinals
[*] Exponentiation of Cardinals
[/LIST]
[*] Sets of Real Numbers
[LIST]
[*] Integers and Rational Numbers
[*] Real Numbers
[*] Topology of the Real Line
[*] Sets of Real Numbers
[*] Borel Sets
[/LIST]
[*] Filters and Ultrafilters
[LIST]
[*] Filters and Ideals 
[*] Ultrafilters
[*] Closed Unbounded and Stationary Sets
[*] Silver's Theorem
[/LIST]
[*] Comblnatorial Set Theory
[LIST]
[*] Ramsey's Theorems
[*] Partition Calculus for Uncountable Cardinals
[*] Trees
[*] Suslin's Problem
[*] Combinatorial Principles
[/LIST]
[*] Large Cardinals
[LIST]
[*] The Measure Problem
[*] Large Cardinals
[/LIST]
[*] The Axiom of Foundation
[LIST]
[*] Well-Founded Relations
[*] Well-Founded Set
[*] Non-Well-Founded Sets
[/LIST]
[*] The Axiomatic Set Theory
[LIST]
[*] The Zermelo-Praenkel Set Theory With Choice
[*] Consistency and Independence
[*] The Universe of Set Theory
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

 

[micromass]

• Author: S.L. Loney
• Title: Plane trigonometry
• Amazon Link: https://www.amazon.com/dp/1418185094/?tag=pfamazon01-20
• Prerequisities: High-School Algebra
• Level: High School

Table of Contents:

[LIST]
[*] Part I
[LIST]
[*] Measurement of angles. Sexagesimal and Centesimal Measure
[*] Circular, or Radian, Measure
[*] Trigonometrical Ratios for angles less than a right angle
[*] Values for angles of 45°, 30°, 60°, 90° and 0°
[*] Simple problems in Heights and Distances
[*] Applications of algebraic signs to Trigonometry
[*] Tracing the changes in the ratios
[*] Trigonometrical ratios of angles of any size. Ratios for -\theta ,90°-\theta,90°+\theta,...
[*] General expressions for all angles having a given trigonometrical ratio 
[*] Ratios of the sum and difference of two angles
[*] Product Formulae 
[*] Ratios of multiple and submultiple angles
[*] Explanation of ambiguities
[*] Angles of 18°, 36°, and 9°
[*] Identities and trigonometrical equations
[*] Logarithms
[*] Tables of logarithms
[*] Principle of Proportional Parts
[*] Sides and Angles of a triangle
[*] Solution of triangles
[*] Given two sides and the included angle
[*] Ambiguous Case
[*] Heights and Distances
[*] Properties of a triangle
[*] The circles connected with a triangle
[*] Orthocentre and Pedal triangle
[*] Centroid and Medians
[*] Quadrilaterals
[*] Regular Polygons
[*] Trigonometrical ratios of small angles. sin \theta < \theta <tan \theta
[*] Area of a Circle
[*] Dip of the horizon
[*] Inverse circular functions
[*] Some simple trigonometrical Series
[*] Elimination
[/LIST]
[*] Analytical Trigonometry
[LIST]
[*] Exponential and Logarithmic Series 
[*] Logarithms to base e 
[*] Two important limits
[*] Complex quantities
[*] De Moivre's Theorem
[*] Binomial Theorem for complex quantities
[*] Expansions of sin n\theta, cos n\theta, and tan n\theta
[*] Expansions of sin a and cos a in a series of ascending powers of a
[*] Sines and Cosines of small angles
[*] Approximation to the root of an equation
[*] Evaluation of indeterminate quantities
[*] Expansions of cos^n \theta and sin^n \theta in cosines or sines of multiples of \theta
[*] Expansions of sin n\theta and cos n\theta in series of descending and ascending powers of sin \theta and cos \theta
[*] Exponential Series for Complex Quantities
[*] Circular functions of complex angles
[*] Euler's exponential values
[*] Hyperbolic Functions
[*] Inverse Circular and Hyperbolic Functions
[*] Logarithms of complex quantities
[*] Value of a^x when a and x are complex
[*] Gregory's Series
[*] Calculation of the value of \pi
[*] Summation of Series 
[*] Expansions in Series 
[*] Factors of x^{2n} - 2x^n cos n\theta + 1 
[*] Factors of x^n - 1 and x^n + 1 
[*] Resolution of sin \theta and cos \theta into factors
[*] sinh \theta and cosh \theta in products
[*] Principle of Proportional Parts
[*] Errors of observation
[*] Miscellaneous Propositions
[*] Solution of a Cubic Equation
[*] Maximum and Minimum Values
[*] Geometrical representation of complex quantities
[*] Miscellaneous Examples
[/LIST]
[/LIST]

 

[micromass]

• Author: S.L. Loney
• Title:The Elements of Coordinate Geometry
• Amazon Link: https://www.amazon.com/dp/B008BCBHDI/?tag=pfamazon01-20
• Prerequisities: High-School Algebra
• Level: High school

Table of Contents:

[LIST]
[*] Introduction. Algebraic Results
[*] Coordinates. Lengths of Straight Lines and Areas of Triangles
[LIST]
[*] Polar Coordinates
[/LIST]
[*] Locus. Equation to a Locus
[*] The Straight Line. Rectangular Coordinates
[LIST]
[*] Straight line through two points
[*] Angle between two given straight lines
[*] Conditions that they may be parallel and perpendicular
[*] Length of a perpendicular
[*] Bisectors of angles
[/LIST]
[*] The Straight Line. Polar Equations and Oblique Coordinates
[LIST]
[*] Equations involving an arbitrary constant
[*] Examples of loci
[/LIST]
[*] Equations representing two or more Straight Lines
[LIST]
[*] Angle between two lines given by one equation
[*] General equation of the second degree
[/LIST]
[*] Transformation of Coordinates
[LIST]
[*] Invariants
[/LIST]
[*] The Circle
[LIST]
[*] Equation to a tangent
[*] Pole and polar
[*] Equation to a circle in polar coordinates
[*] Equation referred to oblique axes
[*] Equations in terms of one variable
[/LIST]
[*] Systems of Circles
[LIST]
[*] Orthogonal circles
[*] Radical axis
[*] Coaxal circles
[/LIST]
[*] Conic Sections. The Parabola
[LIST]
[*] Equation to a tangent
[*] Some properties of the parabola
[*] Pole and polar
[*] Diameters
[*] Equations in terms of one variable
[/LIST]
[*] The Parabola {continued)
[LIST]
[*] Loci connected with the parabola
[*] Three normals passing through a given point
[*] Parabola referred to two tangents as axes
[/LIST]
[*] The Ellipse
[LIST]
[*] Auxiliary circle and eccentric angle
[*] Equation to a tangent
[*] Some properties of the ellipse
[*] Pole and polar
[*] Conjugate diameters
[*] Pour normals through any point
[*] Examples of loci
[/LIST]
[*] The Hyperbola
[LIST]
[*] Asymptotes
[*] Equation referred to the asymptotes as axes
[*] One variable. Examples
[/LIST]
[*] Polar Equation to a Conic
[LIST]
[*] Polar equation to a tangent, polar, and normal
[/LIST]
[*] General Equation. Tracing of Curves
[LIST]
[*] Particular cases of conic sections
[*] Transformation of equation to centre as origin
[*] Equation to asymptotes
[*] Tracing a parabola
[*] Tracing a central conic
[*] Eccentricity and foci of general conic
[/LIST]
[*] General Equation
[LIST]
[*] Tangent
[*] Conjugate diameters
[*] Conics through the intersections of two conics
[*] The equation S=\lambda uv 
[*] General equation to the pair of tangents drawn from any point
[*] The director circle
[*] The foci
[*] The axes
[*] Lengths of straight lines drawn in given directions to meet the onic
[*] Conics passing through four points
[*] Conics touching four lines
[*] The conic LM=R^2
[/LIST]
[*] Miscellaneous Propositions
[LIST]
[*] On the four normals from any point to a central conic
[*] Confocal conics
[*] Circles of curvature and contact of the third order
[*] Envelopes
[/LIST]
[*] Answers
[/LIST]

 

[micromass]

• Author: Kenneth Ireland, Michael Rosen
• Title: A Classical Introduction to Modern Number Theory
• Amazon Link: https://www.amazon.com/dp/1441930949/?tag=pfamazon01-20
• Prerequisities:
• Level: Grad

Table of Contents:

[LIST]
[*] Preface
[*] Unique Factorization
[LIST]
[*] Unique Factorization in Z
[*] Unique Factorization in k[x]
[*] Unique Factorization in a Principal Ideal Domain
[*] The Rings Z[i] and Z[\omega]
[/LIST]
[*] Applications of Unique Factorization
[LIST]
[*] Infinitely Many Primes in Z
[*] Some Arithmetic Functions
[*] \sum 1/p Diverges
[*] The Growth of \pi(x)
[/LIST]
[*] Congruence
[LIST]
[*] Elementary Observations
[*] Congruence in Z
[*] The Congruence ax = b(m)
[*] The Chinese Remainder Theorem
[/LIST]
[*] The Structure of U(Z/nZ)
[LIST]
[*] Primitive Roots and the Group Structure of U(Z/nZ)
[*] nth Power Residues
[/LIST]
[*] Quadratic Reciprocity
[LIST]
[*] Quadratic Residues
[*] Law of Quadratic Reciprocity
[*] A Proof of the Law of Quadratic Reciprocity
[/LIST]
[*] Quadratic Gauss Sums
[LIST]
[*] Algebraic Numbers and Algebraic Integers
[*] The Quadratic Character of 2
[*] Quadratic Gauss Sums
[*] The Sign of the Quadratic Gauss Sum
[/LIST]
[*] Finite Fields
[LIST]
[*] Basic Properties of Finite Fields
[*] The Existence of Finite Fields
[*] An Application to Quadratic Residues
[/LIST]
[*] Gauss and Jacobi Sums
[LIST]
[*] Multiplicative Characters
[*] Gauss Sums
[*] Jacobi Sums
[*] The Equation x^n + y^n = 1 in F_p
[*] More on Jacobi Sums
[*] Applications
[*] A General Theorem
[/LIST]
[*] Cubic and Biquadratic Reciprocity
[LIST]
[*] The Ring Z[\omega]
[*] Residue Class Rings 
[*] Cubic Residue Character
[*] Proof of the Law of Cubic Reciprocity
[*] Another Proof of the Law of Cubic Reciprocity
[*] The Cubic Character of 2 
[*] Biquadratic Reciprocity: Preliminaries
[*] The Quartic Residue Symbol
[*] The Law of Biquadratic Reciprocity
[*] Rational Biquadratic Reciprocity
[*] The Constructibility of Regular Polygons
[*] Cubic Gauss Sums and the Problem of Kummer
[/LIST]
[*] Equations over Finite Fields 
[LIST]
[*] Affine Space, Projective Space, and Polynomials
[*] Chevalley's Theorem
[*] Gauss and Jacobi Sums over Finite Fields
[/LIST]
[*] The Zeta Function
[LIST]
[*] The Zeta Function of a Projective Hypersurface
[*] Trace and Norm in Finite Fields
[*] The Rationality of the Zeta Function Associated to a_0x_0^m + a_1x_1^m + ... + a_nx_n^m
[*] A Proof of the Hasse-Davenport Relation
[*] The Last Entry
[/LIST]
[*] Algebraic Number Theory
[LIST]
[*] Algebraic Preliminaries
[*] Unique Factorization in Algebraic Number Fields
[*] Ramification and Degree
[/LIST]
[*] Quadratic and Cyclotomic Fields
[LIST]
[*] Quadratic Number Fields
[*] Cyclotomic Fields
[*] Quadratic Reciprocity Revisited
[/LIST]
[*] The Stickelberger Relation and the Eisenstein Reciprocity Law
[LIST]
[*] The Norm of an Ideal
[*] The Power Residue Symbol
[*] The Stickelberger Relation
[*] The Proof of the Stickelberger Relation
[*] The Proof of the Eisenstein Reciprocity Law
[*] Three Applications
[/LIST]
[*] Bernoulli Numbers
[LIST]
[*] Bernoulli Numbers; Definitions and Applications
[*] Congruences Involving Bernoulli Numbers
[*] Herbrand's Theorem
[/LIST]
[*] Dirichlet L-functions
[LIST]
[*] The Zeta Function
[*] A Special Case
[*] Dirichlet Characters
[*] Dirichlet L-functions
[*] The Key Step
[*] Evaluating L(s,\chi) at Negative Integers
[/LIST]
[*] Diophantine Equations
[LIST]
[*] Generalities and First Examples
[*] The Method of Descent
[*] Legendre's Theorem
[*] Sophie Germain's Theorem
[*] Pell's Equation
[*] Sums of Two Squares
[*] Sums of Four Squares
[*] The Fermat Equation: Exponent 3
[*] Cubic Curves with Infinitely Many Rational Points
[*] The Equation y^2 = x^3 + k
[*] The First Case of Fermat's Conjecture for Regular Exponent
[*] Diophantine Equations and Diophantine Approximation
[/LIST]
[*] Elliptic Curves
[LIST]
[*] Generalities
[*] Local and Global Zeta Functions of an Elliptic Curve 
[*] y^2 = x^3 + D, the Local Case 
[*] y^2 = x^3 - Dx, the Local Case
[*] Hecke L-functions
[*] y^2 = x^3 - Dx, the Global Case
[*] y^2 = x^3 + D, the Global Case 
[*] Final Remarks
[/LIST]
[*] The Mordell-Weil Theorem
[LIST]
[*] The Addition Law and Several Identities
[*] The Group E/2E
[*] The Weak Dirichlet Unit Theorem 
[*] The Weak Mordell-Weil Theorem 
[*] The Descent Argument
[/LIST]
[*] New Progress in Arithmetic Geometry
[LIST]
[*] The Mordell Conjecture
[*] Elliptic Curves
[*] Modular Curves
[*] Heights and the Height Regulator
[*] New Results on the Birch-Swinnerton-Dyer Conjecture
[*] Applications to Gauss's Class Number Conjecture
[/LIST]
[*] Selected Hints for the Exercises
[*] Bibliography
[*] Index
[/LIST]

 

[micromass]

• Author: Irving Kaplansky
• Title: Fields and Rings
• Amazon link https://www.amazon.com/dp/0226424510/?tag=pfamazon01-20

Table of Contents:

[LIST]
[*] Preface
[*] Fields
[LIST]
[*] Introduction
[*] Field extensions
[*] Ruler and compass construction
[*] Foundations of Galois theory 
[*] Normality and stability
[*] Splitting fields
[*] Radical extensions
[*] The trace and norm theorems
[*] Finite fields
[*] Simple extensions
[*] Cubic and quartic equations
[*] Separability
[*] Miscellaneous results on radical extensions
[*] Infinite algebraic extensions
[/LIST]
[*] Rings
[LIST]
[*] Introduction
[*] The radical
[*] Primitive rings and the density theorem
[*] Semi-simple rings
[*] The Wedderburn principal theorem
[*] Theorems of Hopkins and Levitzki
[*] Primitive rings with minimal ideals and dual vector spaces
[*] Simple rings
[LIST]
[*] The enveloping ring and the centroid
[*] Tensor products
[*] Maximal subfields
[*] Polynomial identities
[*] extension of isomorphisms
[/LIST]
[/LIST]
[*] Homological Dimension
[LIST]
[*] Introduction
[*] Dimension of modules
[*] Global dimension
[*] First theorem on change of rings
[*] Polynomial rings
[*] Second theorem on change of rings
[*] Third theorem on change of rings
[*] Localization
[*] Preliminary lemmas
[*] A regular ring has finite global dimension
[*] A local ring of finite global dimension is regular
[*] lnjective modules
[*] The group of homomorphisms
[*] The vanishing of Ext
[*] lnjective dimension
[/LIST]
[*] Notes
[*] Index
[/LIST]