(From Algebraic Equations to Modern Algebra)

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**About this book :- **
**A History of Abstract Algebra ** written by
** Jeremy Gray **.

This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each. Number Theory: Induction; Binomial Coefficients; Greatest Common Divisors; The Fundamental Theorem of Arithmetic Congruences; Dates and Days. Groups I: Some Set Theory; Permutations; Groups; Subgroups and Lagrange's Theorem; Homomorphisms; Quotient Groups; Group Actions; Counting with Groups. Commutative Rings I: First Properties; Fields; Polynomials; Homomorphisms; Greatest Common Divisors; Unique Factorization; Irreducibility; Quotient Rings and Finite Fields; Officers, Magic, Fertilizer, and Horizons. Linear Algebra: Vector Spaces; Euclidean Constructions; Linear Transformations; Determinants; Codes; Canonical Forms. Fields: Classical Formulas; Insolvability of the General Quintic; Epilog. Groups II: Finite Abelian Groups; The Sylow Theorems; Ornamental Symmetry.
Commutative Rings III: Prime Ideals and Maximal Ideas; Unique Factorization; Noetherian Rings; Varieties; Grobner Bases.

**Book Detail :- **
** Title: ** A History of Abstract Algebra
** Edition: **
** Author(s): ** Jeremy Gray
** Publisher: **
** Series: **
** Year: **
** Pages: ** 412
** Type: ** PDF
** Language: ** Englsih
** ISBN: **
** Country: ** UK

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**About Author :- **
** Jeremy Gray**, School of Mathematics and Statistics, The Open University, Milton Keynes, UK
and Mathematics Institute, University of Warwick, Coventry, UK.

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**Book Contents :- **
**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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