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A History of Abstract Algebra by Jeremy Gray

From Algebraic Equations to Modern Algebra



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A History of Abstract Algebra written by Jeremy Gray, School of Mathematics and Statistics, The Open University, Milton Keynes, UK and Mathematics Institute, University of Warwick, Coventry, UK.

A History of Abstract Algebra written by Jeremy Gray cover the following topics.

  • 1. Simple Quadratic Forms
    1.1 Introduction
    1.2 Sums of Squares
    1.3 Pell’s Equation
    1.4 Exercises

  • 2. Fermat’s Last Theorem
    2.1 Introduction.
    2.2 Fermat’s Proof of the Theorem in the Case n = 4
    2.3 Euler and x3 + y3 = z3
    2.4 Exercises

  • 3. Lagrange’s Theory of Quadratic Forms
    3.1 Introduction
    3.2 The Beginnings of a General Theory of Quadratic Forms
    3.3 The Theorem of Quadratic Reciprocity
    3.4 Exercises
    3.5 Taking Stock

  • 4. Gauss’s Disquisitiones Arithmeticae
    4.1 Introduction
    4.2 The Disquisitiones Arithmeticae and Its Importance
    4.3 Modular Arithmetic
    4.4 Gauss on Congruences of the Second Degree
    4.5 Gauss’s Theory of Quadratic Forms
    4.6 Exercises

  • 5. Cyclotomy
    5.1 Introduction
    5.2 The Case p = 7
    5.3 The Case p = 19
    5.4 Exercises

  • 6. Two of Gauss’s Proofs of Quadratic Reciprocity
    6.1 Introduction
    6.2 Composition and Quadratic Reciprocity
    6.3 Smith’s Commentary on Gauss’s Sixth Proof
    6.4 Exercises

  • 7. Dirichlet’s Lectures on Quadratic Forms
    7.1 Introduction.
    7.2 Gauss’s Third Proof of Quadratic Reciprocity
    7.3 Dirichlet’s Theory of Quadratic Forms
    7.4 Taking Stock

  • 8. Is the Quintic Unsolvable?
    8.1 Introduction
    8.2 Solution of Equations of Low Degree
    8.3 Lagrange (1770)
    8.4 Exercises
    8.5 Revision on the Solution of Equations by Radicals

  • 9. The Unsolvability of the Quintic
    9.1 Introduction.
    9.2 Ruffini’s Contributions
    9.3 Abel’s Work
    9.4 Wantzel on Two Classical Problems
    9.5 Wantzel on the Irreducible Case of the Cubic
    9.6 Exercises

  • 10. Galois’s Theory
    10.1 Introduction
    10.2 Galois’s 1st Memoir
    10.3 From Galois’s Letter to Chevalier
    10.4 Exercises
    10.5 A Cayley Table of a Normal Subgroup.
    10.6 Galois: Then, and Later

  • 11. After Galois
    11.1 Introduction
    11.2 The Publication of Galois’s Work
    11.3 Serret’s Cours d’Algèbre Supérieure
    11.4 Galois Theory in Germany: Kronecker and Dedekind

  • 12. Revision and First Assignment

  • 13. Jordan’s Traité
    13.1 Introduction
    13.2 Early Group Theory: Introduction
    13.3 Jordan’s Traité
    13.4 Jordan’s Galois Theory
    13.5 The Cubic and Quartic Equations

  • 14. The Galois Theory of Hermite, Jordan and Klein
    14.1 Introduction
    14.2 How to Solve the Quintic Equation
    14.3 Jordan’s Alternative
    14.4 Klein
    14.5 Klein in the 1870s
    14.6 Klein’s Icosahedron
    14.7 Exercises

  • 15. What Is ‘Galois Theory’?
    15.1 Introduction
    15.2 Klein’s Influence
    15.3 Concluding Remarks

  • 16. Algebraic Number Theory: Cyclotomy
    16.1 Introduction
    16.2 Kummer’s Cyclotomic Integers
    16.3 Fermat’s Last Theorem in Paris

  • 17. Dedekind’s First Theory of Ideals
    17.1 Introduction
    17.2 Divisibility and Primality
    17.3 Rings, Ideals, and Algebraic Integers
    17.4 Dedekind’s Theory in 1871

  • 18. Dedekind’s Later Theory of Ideals
    18.1 Introduction
    18.2 The Multiplicative Theory
    18.3 Dedekind and ‘Modern Mathematics’
    18.4 Exercises

  • 19. Quadratic Forms and Ideals
    19.1 Introduction
    19.2 Dedekind’s 11th Supplement, 1871–1894
    19.3 An Example of Equivalent Ideals

  • 20. Kronecker’s Algebraic Number Theory
    20.1 Introduction
    20.2 Kronecker’s Vision of Mathematics
    20.3 Kronecker’s Lectures
    20.4 Gyula (Julius) König

  • 21. Revision and Second Assignment

  • 22. Algebra at the End of the Nineteenth Century
    22.1 Introduction
    22.2 HeinrichWeber and His Textbook of Algebra
    22.3 Galois Theory
    22.4 Number Theory

  • 23. The Concept of an Abstract Field
    23.1 Introduction
    23.2 Moore, Dickson, and Galois Fields
    23.3 Dedekind’s 11th Supplement, 1894
    23.4 Kürschák and Hadamard
    23.5 Steinitz

  • 24. Ideal Theory and Algebraic Curves
    24.1 Introduction
    24.2 The Brill–Noether Theorem
    24.3 The Failure of the Brill–Noether Theorem to Generalise
    24.4 Lasker’s Theory of Primary Ideals
    24.5 Macaulay’s Example
    24.6 Prime and Primary Ideals

  • 25. Invariant Theory and Polynomial Rings
    25.1 Introduction
    25.2 Hilbert
    25.3 Invariants and Covariants
    25.4 From Hilbert’s Paper on Invariant Theory (1890)
    25.5 The Hilbert Basis Theorem and the Nullstellensatz

  • 26. Hilbert’s Zahlbericht
    26.1 Introduction
    26.2 An Overview of the Zahlbericht
    26.3 Ideal Classes and Quadratic Number Fields
    26.4 Glimpses of the Influences of the Zahlbericht

  • 27. The Rise of Modern Algebra: Group Theory
    27.1 Introduction
    27.2 The Emergence of Group Theory as an Independent Branch of Algebra
    27.3 Dickson’s Classification of Finite Simple Groups

  • 28. Emmy Noether
    28.1 Introduction
    28.2 Ideal Theory in Ring Domains
    28.3 Structural Thinking

  • 29. FromWeber to van derWaerden
    29.1 Introduction
    29.2 van der Waerden on the Origins of Moderne Algebra

  • 30. Revision and Final Assignment
    A Polynomial Equations in the Eighteenth Century
    A.1 Introduction
    A.2 The Fundamental Theorem of Algebra Before Gauss
    B Gauss and Composition of Forms
    B.1 Composition Theory
    B.2 Gaussian Composition of Forms
    B.3 Dirichlet on Composition of Forms
    B.4 Kummer’s Observations
    C Gauss’s Fourth and Sixth Proofs of Quadratic Reciprocity
    C.1 Gauss’s Fourth Proof
    C.2 Gauss’s Sixth Proof
    C.3 Commentary
    D From Jordan’s Traité
    D.1 Jordan, Preface to the Traité
    D.2 Jordan, General Theory of Irrationals
    D.3 Jordan: The Quintic Is Not Solvable by Radicals
    D.4 Netto’s Review
    E Klein’s Erlanger Programm, Groups and Geometry
    E.1 Introduction
    E.2 Felix Klein
    E.3 Geometric Groups: The Icosahedral Group
    E.4 The Icosahedral Equation
    F From Dedekind’s 11th Supplement (1894)
    G Subgroups of S4 and S5
    G.1 The Subgroups of S4
    G.2 The Subgroups of S5
    H Curves and Projective Space
    H.1 Intersections and Multiplicities
    I Resultants
    I.1 Netto’s Theorem
    I.2 Resultants
    j FurtherReading
    J.1 Other Accounts of the History of Galois Theory
    J.2 Other Books on the History of Algebraic Number Theory
    References
    Index

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