A History of Abstract Algebra

Frontmatter

Chapter 1. Simple Quadratic Forms

Abstract

In this chapter we look at two topics: the question of what numbers, and specifically what primes, can be written in the form x ² ± ny ² for small, non-square positive n?, and how to show that the equation x ² − Ay ² = 1 has solutions in integers for positive, non-square integers A. Once we have seen what mathematical conclusions we shall need, we look at how to handle the mathematics in a historical way—how to use mathematics as evidence.

Jeremy Gray

Chapter 2. Fermat’s Last Theorem

Abstract

In this chapter we look at Fermat’s account of the equation x ⁴ + y ⁴ = z ⁴ and then at Euler’s flawed but insightful account of x ³ + y ³ = z ³.

Jeremy Gray

Chapter 3. Lagrange’s Theory of Quadratic Forms

Abstract

It was Lagrange who sought to produce a general theory of quadratic forms, after Euler had published a number of deep and provocative studies of many examples—what would today be called ‘experimental mathematics’. In this chapter we look at one key idea in his treatment: the reduction of forms to simpler but equivalent ones. We are led to one of the great theorems in mathematics: quadratic reciprocity. It was conjectured well before it was proved for the first time, as we shall see later.

Jeremy Gray

Chapter 4. Gauss’s Disquisitiones Arithmeticae

Abstract

Carl Friedrich Gauss established himself as a mathematician at the age of 24 with the publication of his Disquisitiones Arithmeticae, which eclipsed all previous presentations of number theory and became the standard foundation of future research for a century. At its heart is a massive reworking of the theory of quadratic forms. In this chapter we first present an overview of the work before we concentrate on how this book illuminated the many puzzling complexities still left after Lagrange’s account.

Jeremy Gray

Chapter 5. Cyclotomy

Abstract

Following an overview of Carl Friedrich Gauss’s Disquisitiones Arithmeticae in the previous chapter, in this chapter we turn to another major topic in Gauss’s book: cyclotomy. We will see how Gauss came to a special case of Galois theory and, in particular, to the discovery that the regular 17-sided polygon can be constructed by straight edge and circle alone.

Jeremy Gray

Chapter 6. Two of Gauss’s Proofs of Quadratic Reciprocity

Abstract

In this chapter, we discuss two of Gauss’s proofs of quadratic reciprocity: one (his second) uses composition of forms, and the other (his sixth) uses cyclotomy. The sixth was the last proof of this theorem he published, although he went on to leave two more unpublished.

Jeremy Gray

Chapter 7. Dirichlet’s Lectures on Quadratic Forms

Abstract

Gauss’s first proof of quadratic reciprocity was given in Section IV of the Disquisitiones Arithmeticae. Gauss went on to publish five more, leaving a further two unpublished; he also knew that these different proofs hinted at important connections to as-yet undiscovered parts of mathematics. But insofar as some of these proofs were intended to explore or illustrate these connections they were of varying levels difficulty and not all suitable for beginners. The simplest is the third proof, which was adopted by Peter Gustav Lejeune Dirichlet in lectures that he gave in the 1850s and which formed part of his book Vorlesungen über Zahlentheorie (Lectures on Number Theory) that did so much to bring number theory to a wide audience of mathematicians. We look at this proof here, and then turn to look at Dirichlet’s Lectures more broadly.

Jeremy Gray

Chapter 8. Is the Quintic Unsolvable?

Abstract

In this chapter, we turn to what today is regarded as a different branch of algebra, the solution of polynomial equations, although Gauss’s work on the ‘higher arithmetic’ was not automatically regarded as being part of algebra. We shall find that polynomial algebra also evolved in the direction of deepening conceptual insight, so here we witness again one of the origins of the transformation from school algebra to modern algebra.

Jeremy Gray

Chapter 9. The Unsolvability of the Quintic

Abstract

This chapter is concerned with another momentous advance in algebra around 1800: the discovery that the quantic equation cannot be solved by an algebraic formula. But how can a negative of this kind be proved? What is it to analyse how a problem can be solved? Here, we look at the works of Ruffini and (in particular) Abel after considering what it involves to solve an equation algebraically.

Jeremy Gray

Chapter 10. Galois’s Theory

Abstract

This chapter focuses on Galois’s work. Can it be that although he had done his best to present the complete resolution of the question “When is a polynomial equation solvable by radicals”, what we can see are pieces of this resolution, key pieces presumably, but we don’t find them convincing? Is it that the pieces are not all there, or is there something about them that we cannot see? There is a way of thinking about the problem that Galois presented which we do not follow. Is it because he was too quick? In that case, patient work should enable us to catch up. Or is there some deep insight he has failed to present?

Jeremy Gray

Chapter 11. After Galois

Abstract

Galois’s theory was considered very difficult in its day, and was also poorly published. This chapter looks at what had to happen before it could become mainstream mathematics, and how as it did so it changed ideas about what constitutes algebra and started a move to create a theory of groups.

Jeremy Gray

Chapter 12. Revision and First Assignment

Abstract

In this chapter, we revise the topics explored so far and discuss the first assignment. We do not boil the previous 11 chapters down to a misleading ’essence’. Instead, we raise some questions and let you answer them yourselves—to your own satisfaction.

Jeremy Gray

Chapter 13. Jordan’s Traité

Abstract

The book that established group theory as a subject in its own right in mathematics was the French mathematician Jordan’s Traité des Substitutions et des Équations Algébriques of 1870. In this chapter, we look at what that book contains, and how it defined the subject later known as group theory.

Jeremy Gray

Chapter 14. The Galois Theory of Hermite, Jordan and Klein

Abstract

We now turn to look at how Galois Theory became established. In this chapter, we will look at an alternative approach to the question of solving quantic equations that is associated with the influential figures of Hermite, Kronecker and Brioschi. We then consider the alternative promoted by Jordan and by Felix Klein, whose influence on the development of Galois theory in the late nineteenth century is often neglected, but was in fact decisive.

Jeremy Gray

Chapter 15. What Is ‘Galois Theory’?

Abstract

In this chapter, we are still concerned with the question of how Galois Theory became established as we look at Klein’s influence in more detail.

Jeremy Gray

Chapter 16. Algebraic Number Theory: Cyclotomy

Abstract

In this chapter, we return to one of Gauss’s favourite themes, cyclotomic integers, and look at how they were used by Kummer, one of the leaders of the next generation of German number theorists. French and German mathematicians did not keep up-to-date with each other’s work, and for a brief, exciting moment in Paris in 1847 it looked as if the cyclotomic integers offered a chance to prove Fermat’s last theorem, only for Kummer to report, via Liouville, that problems with the concept of a prime cyclotomic integer wrecked that hope. Primality, however, turned out to be a much more interesting concept, and one of the roots of the concept of an ideal.

Jeremy Gray

Chapter 17. Dedekind’s First Theory of Ideals

Abstract

This chapter picks up from the previous one and looks at how Dedekind analysed the concept of primality in an algebraic number field. This was to mark the start of a sharp difference of opinion with Kronecker.

Jeremy Gray

Chapter 18. Dedekind’s Later Theory of Ideals

Abstract

In this chapter, we look at how Dedekind refined his own theory of ideals in the later 1870s, and then at the contrast with Kronecker

Jeremy Gray

Chapter 19. Quadratic Forms and Ideals

Abstract

One of the successes of Dedekind’s theory was the way it allowed Gauss’s very complicated theory of the composition of quadratic forms to be re-written much more simply in terms of modules and ideals in a quadratic number field, which in turn explained the connection between forms and algebraic numbers. In this chapter, we look at how this was done.

Jeremy Gray

Chapter 20. Kronecker’s Algebraic Number Theory

Abstract

In this chapter, we look at the Kroneckerian alternative to Dedekind’s approach to ‘ring theory’ set out in his Grundzüge and later extended by the Hungarian mathematician Gyula (Julius) König. This leads us to the emergence of the concept of an abstract field.

Jeremy Gray

Chapter 21. Revision and Second Assignment

Abstract

In this chapter, we again revise the topics explored so far. We then discuss the second assignment. As before, we raise some questions and ask you to answer them yourselves, to your own satisfaction.

Jeremy Gray

Chapter 22. Algebra at the End of the Nineteenth Century

Abstract

In this chapter we begin to look at the major change that happened to algebra in the nineteenth century: the transformation from polynomial algebra to modern algebra. A vivid impression of the subject is given by the book that described the state of the art around 1900, Weber’s Lehrbuch der Algebra, much of which described Galois theory and number theory as it then stood.

Jeremy Gray

Chapter 23. The Concept of an Abstract Field

Abstract

In this chapter, we look at three aspects of the theory of abstract fields: the discovery that all finite fields are ‘Galois’ fields; Dedekind’s presentation of the ‘Galois correspondence’ between groups and field extensions; and the emergence of the concept of an abstract field.

Jeremy Gray

Chapter 24. Ideal Theory and Algebraic Curves

Abstract

Polynomials in two variables define algebraic curves in the plane, and algebraic curves in the plane generally meet (perhaps in complicated ways) in points. What is the connection between the geometry and the algebra? In this chapter we shall see how this question was answered, not entirely successfully, in the late nineteenth century by two mathematicians: Alexander Brill and Max Noether (the father of the more illustrious Emmy). The generalisation to more variables was very difficult, and was chiefly the achievement of Emanuel Lasker, who was the World Chess champion at the time, with his theory of primary ideals. We shall give an example of his fundamental result taken from the English mathematician F.S. Macaulay’s fundamental work on polynomial rings. With these results, the basic structural features of polynomial rings were all in place, and with the equally rich theory of number fields so too were all the basic features of commutative algebra.

Jeremy Gray

Chapter 25. Invariant Theory and Polynomial Rings

Abstract

This chapter is concerned with another, related branch of algebra that mingles polynomials with geometry: invariant theory. We shall see how this field was decisively rewritten by the young David Hilbert in work that made his name in the international mathematical community.

Jeremy Gray

Chapter 26. Hilbert’s Zahlbericht

Abstract

Historians often note that two books in number theory open and close the nineteenth century in the theory of numbers: Gauss’s Disquisitiones Arithmeticae at the start of the nineteenth century and Hilbert’s Zahlbericht, or Report on the Theory of Numbers, at the end. In this chapter, we look at aspects of Hilbert’s book, and hint at some of its influential choices, including its influence on the subsequent study of algebra.

Jeremy Gray

Chapter 27. The Rise of Modern Algebra: Group Theory

Abstract

In this chapter, we look at how group theory also emerged “in its own right” in various books and papers around 1900, including the first monograph on abstract group theory, and then look at the influential work of the American mathematician Leonard Eugene Dickson.

Jeremy Gray

Chapter 28. Emmy Noether

Abstract

Emmy Noether is universally regarded as the greatest woman mathematician so far. In this chapter, we look at some of her work in the creation of modern algebra.

Jeremy Gray

Chapter 29. From Weber to van der Waerden

Abstract

This chapter includes a commentary on the opening chapter of Corry’s Modern algebra and the rise of mathematical structures, and more particularly his paper ‘From Algebra (1895) to Moderne Algebra (1930): Changing Conceptions of a Discipline. We also explore van der Waerden’s paper ‘On the sources of my book Moderne Algebra’.

Jeremy Gray

Chapter 30. Revision and Final Assignment

Abstract

This final chapter is devoted to revising the topics and a discussion of the final assignment.

Jeremy Gray

Backmatter