The development of Galois theory

Fiona Brunk

Galois' commentators

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(Évariste Galois)
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The first mathematician who published a commentary on Galois' two main papers was Enrico Betti (1823-1892). In his 1852 publication Sulla risoluzione delle equazioni algebriche, he presented Galois' ideas in a more accessible manner, making a few additions of his own. His main contribution was to fill in gaps in certain proofs by Galois, Abel and Ruffini but he also generalised some of Galois' group theoretical results.

Like his contemporaries, Betti certainly overemphasised the application of Galois Theory to the solvability of equations, and he seemed slightly confused over some group theoretical aspects. For example, he does not start his proofs by defining a group G, but instead normalises in some unspecified universe which leads to a confusion of the group with its quotient groups. However, he achieves some interesting results and advances the understanding of group theory, for example by extending conjugation from elements to subgroups. Furthermore, Betti's paper remained the only discussion of Galois Theory until the publication of Jordan's paper in 1870, and it would have been more widely used had it not been published in Italian. The main mathematical language of the time was French and so the next significant step in the development of Galois Theory was its publication in a French algebra textbook.

Joseph Serret's (1819-1885) Cours d'algèbre supérieure was the main algebra textbook for half a century. The third edition, published in 1866, contained the first exhibition of Galois Theory in a textbook. Serret made a few notational contributions which helped to clarify some ideas but his main significance to the development of Galois Theory was that his presentation reached a wider audience. His understanding of group theory may have been less advanced than Betti's but Serret's clarification and organisation of Galois' Mémoire was crucial and lasting.

Serret's Cours d'algèbre was quickly accepted internationally, and remained the main text on algebra for the next 40 years. It may be argued that this was not beneficial to the development of algebra, particularly in France: Serret was incapacitated by illness after 1871 but his textbook was so popular that new editions of it were published as late as 1928. The later editions did not contain any results which had been published after 1866, and the fact that Cours d'algèbre remained the main algebra reference in France nevertheless greatly slowed down French research progress in algebra, including Galois Theory.

Something more positive which can be said about the Cours d'algèbre is that it inspired Camille Jordan (1838-1922) to write his Traité des substitutions et des équations algébrique which Kiernan [3] describes as the most important French publication on algebra in the second half of the 19th century. Jordan had made several publications on group theory in the 1860s. In the Traité, published in 1870, he compiled everything that was known on group theory at the time. Throughout this text, a group was still connected with permutations and algebraic equations. Nonetheless, the Traité was the first paper whose central object of study was the group which is why Jordan is often seen as the first modern algebraist. As other commentators before him, he filled in gaps in some of Galois' proofs and added some results of his own to his presentation of Galois Theory. However, he went further than Betti and Serret by truly reformulating many of Galois' results to better suit his approach which viewed the group as central. After presenting Jordan's formulation of Galois Theory, Kiernan [3] comments [3]:
This is no longer just a theory of equations; these are theorems about groups, whose results are applicable to equations and their solution.

For the remainder of the 19th century, most progress in Galois Theory was due to the development of Field Theory in Germany. While Lagrange had induced the beginnings of group theory in France, Gauss had been the main influence on mathematics in Germany. His results in number theory inspired mathematicians such as Kronecker and Dedekind to develop what is known as Field Theory today. Although Field Theory was already quite advanced before Galois Theory was closely associated with it, both Kronecker and Dedekind contributed to the development of Galois Theory. For instance, Kronecker was first to describe the Galois group not in terms of permutations on the roots of an equation, but as a group of automorphisms of the coefficient field with adjoined quantities. Dedekind's significance was largely due to the essential progress he made in Field Theory, i.e. many of his results were fundamental to Artin's later formulation of Galois Theory. Other German writers of the time such as Felix Klein, Eugen Netto and Walther von Dyck made important advances in group theory. The last significant group theoretical gap in Galois Theory was closed in 1889 when Otto Hölder (1859-1937) proved what is known as the Jordan-Hölder Theorem today.

So the end of the 19th century was an active period in the development of Galois Theory: The group theorists of the time filled in the last remaining gaps in Galois' proofs which completed the development of classical Galois Theory. The field theorists of the time developed the foundations of the last major reformulation of Galois Theory to this day which was completed by Artin. There were some important expository papers around 1900 which will be discussed shortly, but Galois Theory itself remained largely unaltered from about 1900 until the 1930s.

The first German presentation of Galois Theory was Paul Bachmann's 1881 article Über Galois' Theorie der algebraischen Gleichungen. After a German translation of Galois' works was published in 1889, several reviews appeared in the 1890s. Most important among these were the discussions by Heinrich Weber (1842-1913), namely his article Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie, published in 1893, and the section on Galois Theory in Weber's 1895 textbook Lehrbuch der Algebra. Weber presented Galois Theory in terms of group theory and field theory, making very few references to equations, so that the theory could also be applied to other areas than the solvability of equations. Weber's theorems are no longer restricted to the rationals, but apply to arbitrary fields.

Many Galois Theory concepts seem very complicated in Weber's notation because he is very formal and perhaps too careful in his distinctions. However, he finally combines field theory and group theory in Galois Theory in a consistent way. Furthermore, he is the first author to re-establish the distinction between Galois Theory and its applications which Galois had in mind. The chapter on Galois Theory in Weber's algebra textbook makes no reference to the solvability of groups or the solvability of equations by radicals. These topics are dealt with in the following chapter, Application of permutation groups to equations, although even here, Weber seems more interested in the properties of specific groups than in the application of these properties to the solvability of equations.

Weber's article and textbook are the first modern discussions of Galois Theory. They are presented as the study of field extensions and their automorphism groups. Even when Galois Theory is applied to the solvability of equations, the nature of the solution is the object of study. The aim is no longer to find a procedure for determining the solutions of a given Weber. Hence Weber's treatment was well ahead of his time. The first mathematician after Weber to deal with Galois Theory at such an advanced level was Emil Artin 40 years later.

Weber's publications were not the only presentations of Galois Theory around 1900. For instance, the first English expositions of Galois Theory were Oskar Bolza's 1891 article On the Theory of Substitution-Groups and its Application to Algebraic Equations and the 1892 translation of Netto's Substitutionentheorie. In these papers, we can detect some interest in the structure of the Galois group but they do not go far beyond Betti's presentation of Galois Theory. An even less advanced, but very popular exposition of Galois Theory was James Pierpont's Galois' Theory of Algebraic Equations, published in 1899/1900. While his approach was existential, his main concern was the constructability of the group of an equation and the problems arising from this, and his methods were highly computational. To give another example of the variety of approaches to Galois Theory around 1900, Henri Vogt's approach was even more computational than Pierpont's. Where even Pierpont merely proves that construction of the Galois group is possible, Vogt actually constructs it. His 1895 paper Leçons sur la résolution algébrique des équations was essentially a manual for solving equations which had very little to do with Galois Theory as we understand it today.

Hölder's article Galois'sche Theorie mit Anwendungen in the 1898 Enzyklopädie der Mathematischen Wissenschaften was perhaps more representative of the general understanding of Galois Theory around 1900. Hölder does not present Weber's major innovations but he follows Weber in separating the theory from its applications. Such was the understanding of Galois Theory among advanced mathematicians around 1900. By the early 20th century, Galois Theory was considered a finished subject and active mathematical research moved on to other areas.

Yet the last major step towards today's understanding of Galois Theory was achieved in two sets of lecture notes by Emil Artin (1898-1962): Foundations of Galois Theory, published in 1938, and Galois Theory, published in 1942. According to Artin, Galois Theory studies how field extensions are related to their automorphism groups. He wanted to present the theory independently of its application to the solution of equations. While field theory had been used before to simplify the presentation of Galois Theory, the structure of fields had been largely ignored. Artin abandoned the approach of building a sequence of field extensions by adjoining successive resolvents to the coefficient field. Instead, his starting point was to consider the splitting field of the equation (which is the smallest field containing the roots and coefficients). The splitting field existed by Kronecker; Artin did not investigate the question of how it could be constructed in practice. Thus the historic application of Galois Theory was finally reduced to the question of whether there existed an extension of the coefficient field containing the splitting field. Previous mathematicians, such as Bartel Leendert van der Waerden in Moderne Algebra (1930), had incorporated modern ideas from Linear Algebra and Field Theory into Galois Theory, but Artin was first to make use of the crucial idea of the splitting field in this way. His approach was heavily influenced by Dedekind, Kronecker and Weber, but Artin was able to comprehend how it all fitted together and gave a very succinct, precise and modern presentation of Galois Theory.

Artin gathers together all important conclusions in one Fundamental Theorem of Galois Theory [3]:
If p(x) is a separable polynomial over a field F, and G the group of the equation p(x) = 0 where E is the splitting field of p(x), then:
  1. Each intermediate field B is the fixed field for a subgroup GB of G, and distinct subgroups have distinct fixed fields.
  2. The intermediate field B is a normal extension of F if and only if the subgroup GB is a normal subgroup of G. In this case the group of automorphisms of B which leave F fixed is isomorphic to the quotient group G/GB.
  3. For each intermediate field B, [B:F] is equal to the index of GB and [E:B] is equal to the order of GB.

This Fundamental Theorem makes no reference to substitutions of roots; it talks about field extensions and their automorphism groups. A polynomial is mentioned only to produce a splitting field in relation to the ground field -- Galois Theory is no longer about polynomials. Today's formulations (see [6] or [7]) of the Fundamental Theorem of Galois Theory are equivalent to Artin's; their aim is to reveal the parallel structure of the extension field and its automorphism group.

Artin followed Weber in making a clear distinction between theory and application. Solvable groups and the idea of solvability by radicals appear only in an appendix to his text, and only for historic reasons. In the 1930s, mathematicians were certainly much more receptive to abstract algebra than Weber's audience had been, but Artin's crucial innovation was the way in which he combined results by Dedekind, Kronecker, Weber and others in order to present a new conception of Galois Theory. In short, Artin unified and completed all the earlier approaches in a formulation of Galois Theory which we still use today.

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