# How did the modern understanding of Galois theory come about?

The "modern" understanding of the Galois group of a polynomial is as automorphisms of the splitting field of the polynomial which keep the base field fixed. These concepts were unknown to Galois, who thought about permutations of roots and "substitutions." These formulations are entirely equivalent.

The question is: where did the modern formulation come from? It seems likely that it was already in use in the course given by Noether and Artin in the 1920's, on which van der Waerden's Moderne Algebra was based. (Certainly this is the point of view in the 7th German edition). And it was presumably already well in place by the time Artin's 1948 lectures on Galois theory appeared. But maybe its origin was earlier, with Kronecker, or Dedekind, or Kummer?

• It seems that it comes from Noether and Artin. Even the modern definition of a group was only stated in 20th century. Dec 24 '18 at 18:45
• Yes, although you would not necessarily need a formal definition of group to be able to talk about automorphisms. Dec 24 '18 at 21:49
• @Conifold I hope you don't mind if I edit your edit of my title so that it refers to "Galois theory" rather than "the Galois theory." Jan 11 '19 at 16:23

The systematic modern terminology and presentation of the Galois theory is due to Artin, a part of his joint project with Emmy Noether to reformulate the "concrete" older algebra in abstract terms, inspired by Dedekind. It was Artin who finally detached the Galois theory from the problem of solving equations algebraically, and gave a presentation that freely moves between considering the same object as a field, as a group, or as a vector space, as needed.

However, a book length presentation of the prior history in Kiernan's The development of Galois theory from Lagrange to Artin traces the ideas behind it to Kronecker, Dedekind and Weber. Kronecker addressed many points of Galois theory in a new light using his domain of rationality concept (not quite the modern field, but close). His treatment was developed in a number of papers since 1853, culminating in Grundzuge einer arithmetischen Theorie der algebraischen Gröβen (1882, Jen. Werke, II, 237-387). Here is Kiernan, p.127:

"Kronecker makes reference in this article to many points of Galois' work. He is primarily concerned with the concept of adjunction alone; so this is not a development of Galois Theory. Nevertheless he does provide a concept of a system, the domain of rationality of the coefficients, to which the roots of the successive auxiliary equations may be adjoined. There can also be recognized, in very primitive fashion, the connection between the group of the equation and the automorphisms on the extension fields of the field of coefficients. Of course, Kronecker could not express the result in this way, since he did not see a domain of rationality as a completed entity, but merely as a region or place which contained the result of any finite number of algebraic operations on its elements, or rather where these operations took place."

Dedekind lectured on the Galois theory at Göttingen in 1856-57, the same time he corresponded with Kronecker about it, and introduced the concept of a field (rational Gebiet at the time, Körper referred only to complex number fields then) in 1857-58. His treatment became standard in German universities for a while, but he reworked it anew in the 1890-s.

"In the 1894 edition of the Zahlentheorie, Dedekind developed a theory of field extensions which is in some ways similar to that of Kronecker. In his development, Dedekind specifically refers to Galois and his work on the concept of adjoining elements to the set of coefficients of an equation… Dedekind then develops the idea of a field isomorphism, which he calls a "permutation of the field"... Further, he recognizes that any "permutation" $$\phi$$ of a field $$K$$ is an identity mapping on $$Q$$, that is, $$\phi$$ maps each element of the rational field $$Q$$ onto itself, no matter what it does to the other elements of $$K$$. Also, if $$\phi$$ is a permutation, that is, an isomorphism, of the field K onto K', then, says Dedekind, the set of elements in K left fixed by $$\phi$$ forms a field containing $$Q$$, called the field belonging to $$\phi$$... Many results developed here by Dedekind on the interpretation of an extension field as a vector space over the ground field were later used by Artin in his formulation of Galois Theory… In retrospect it is surprising how little immediate use was made of the work either of Kronecker or of Dedekind in field theory. The modern development of field theory, so brilliantly begun by Dedekind, lay largely dormant for another 30 years." [Kiernan, p.128ff]

To the extent that the field concept was used for the Galois theory at all in the late 19th century, it was Kronecker's rather than Dedekind's. One exception to this trend was Weber in his Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie (1893), and Lehrbuch der Algebra (1895):

"Weber's development of Galois Theory is certainly the first modern treatment of the topic. The development is not restricted to the rational field, but to an arbitrary field. The theory itself is clearly acknowledged as concerned only with the extensions of the ground field and the groups of automorphisms of these extensions. Even the consideration of the process of solution is made secondary to the study of the nature of the solution. But for a paper to be regarded as modern today, it must necessarily have been years ahead of its time in the 1890's. As will be seen in the next section, Weber's presentation stands almost alone in its own generation. It may be viewed as a direct predecessor of Artin's work, forty years later." [Kiernan, p.141]

• Thank you, that is wonderful! I am glad to know about Kiernan's book. Based on the quotes above, it seems that while Weber's presentation was not entirely "modern," the idea of the Galois group as field automorphisms may in fact be his. Dec 26 '18 at 2:09
• @NatKuhn It appears also in Dedekind's Zahlentheorie (I added the relevant sentences to the second quote), and they likely arrived at it independently. Dedekind might have come up with it before Weber since he worked on Galois theory long before 1894. I am also not sure if Artin was familiar with Weber's work, but he was certainly familiar with Dedekind's. Dec 26 '18 at 9:22
• Thank you very much, that is very helpful, @Conifold! Dec 27 '18 at 1:38
• @Conifold Why do tou think they were developed independently? Dedekind and Weber were good friends and collaborated closely on their work on algebraic functions, so it wouldn't surprise me if they knew of each other's work... Dec 28 '18 at 20:06
• @Nagase That is true, and Weber acknowledges Dedekind's influence in his 1893 paper. But he refers to Dedekind's older work on Galois theory, and uses a different terminology for field automorphisms (Dedekind called them "permutations"). In the Lehrbook Weber calls them "substitutions", and does not refer to the fact that they are automorphisms explicitly, when he proves that they form a group. Dec 30 '18 at 3:36

My impression is that Artin's role in the development of Galois theory is usually greatly overrated. Kiernan's article jumps from 1900 to Artin in the late 1930s. Certainly a "modern" Galois theory existed during the intervening years --- see for example, Albert's Algebra books. Artin's main contribution was to give a proof of the main theorem of Galois theory without using the existence of a primitive element. Certainly choosing a primitive element is unnatural, but I don't think Artin's proofs are any easier for students than the older proofs (as in Albert).