La theorie de Galois C19th — Part 2

0:00 All of this is covered in another chapter, not in the Javi-Claurite-Gallois book. On the other hand, this is a translation for French, but if you understand what I'm saying, what strikes me today is that it is extremely modern. It seems extremely modern. On the other hand, Tiret is a good guy who didn't go to school. How is the concept of the body presented? In an axiomatic way or not? The concept of the body? No, not yet. But what is interesting is that it is not axiomatic in the sense that we presented the groups axiomatically. So it is not a structural analysis. But despite everything, what is interesting about it is that the concept of the body is central. Really central. It's not substitutions, for example, that are central here. Not at all. So, you see that this is the first time, in fact, where we see reappear, in fact, what we have characterized as the Galois approach, this principle of dialogue between attention, the diminution of the web, etc. And so, if we sum up, in fact... What appears in this manual is that we have conceptual boundaries, conceptual boundaries that are quite new for the Galois theory, that is to say that in the end, it is the theoretical aspects that are written in the Galois theory, the aspects of the notion of adjunction, the notion of the body, a little bit the substitutions, even if for him it is something that is annexed ... But on the other hand, all the aspects that concern the resolution of the equations, to put it bluntly, are not part of Galois' theory. The aspects that concern the resolution, in particular the question of the reduction of the resolution, etc., are treated elsewhere than in the Galois' theory chapter.

2:30 The first thing is that, indeed, Wilbert's law, when we look at it, is much more familiar with this kind of thing today than with the work of Netto, which seems a bit obscure when we look at it. So, in my opinion, it is essentially linked to the fact that this is a vision of algebra which is derived directly from the rule of Elkin, which corresponds to the one that was finally imposed in the 19th century. It must be kept in mind that at the time when these works were written, both Netto and Weber represented alternatives that were also taken seriously at the beginning of the 19th century. In particular, the two works that I will show you right after, the two French works, put on the same level Netto's manual and Weber's manual. One is better than the other, one is more legitimate than the other. Netto's manual was a great success in the beginning of the 20th century. Now, the other book is much more famous, but Netto was quite legitimate. He wrote many translations. This also means that it was quite conceivable at the end of the 19th century to build a theory of equations that was a modern theory of equations. With modern tools, which are far beyond what we know, but which at the same time are not a theory of Gallois. And for this, the research of Kronecker, in particular, constituted a very solid alternative and which was very seriously taken into consideration. Finally, it is only after 1900... The idea is that Lepkin appears as a master of thought in a way and that in the end we begin to use more Lepkin and less Kroeniger, but in fact for the contemporaries, Kroeniger even had an institutional position much stronger than that of Lepkin, he was perhaps more recognized.

5:00 This is a retrospective reading after which we can see that at the beginning, one version or the other was also very clear. The other remark that I would like to make here is that these books, when we compare them, always show very clearly the difference in the systemological culture that there could be between the mathematics in Gottingen and the mathematics in Berlin at the same time. In my opinion, there are in fact in competition two ways to envisage and to make algebra, and on this I find that precisely this type of source is quite interesting to make this kind of culture appear, since we can consider them as workers who make quite visible the different cultures that are at stake. The question I would like to ask you now is if we move to France at the beginning of the 20th century, what is the Galois theory in France at the beginning of the 20th century? So, first of all, if you look at all the sources that I have listed on the Galois, in particular my part of Galois in France, The main sources at the end of the 19th century and the beginning of the 20th century are the manual of Serret and the Treaty of the Substitution of Journals. These two books constitute the basis of the theory of law in France at the end of the 19th century. One will be at the level of student training and the other will be considered as a more advanced level. For example, if we look at a journal called the Nouvelle Zonale de Mathématiques, which is a journal dedicated to students who have to prepare, or students who don't have to prepare, there is only one article for the 39e published in 1888, and this article is written by Dolmien. In fact, what he retains from the Galois theory is a unique article.

7:30 All of these terms refer explicitly to the Galois theory. What is noteworthy is that it is precisely the only criterion for resolution, and it is an article that is based on the practices put in place in the book of these researches. It is an article for which the Galois theory is in fact part of the theory of equations which serves to achieve... So, to give you a little bit of detail, in fact, Domien re-demonstrates the Galois theorem by using, to the maximum, elementary knowledge on substitutes. It is placed a little in the same... It is a little in the same spirit as the demonstration we saw of Hermite earlier. It is a demonstration that does not use a conceptual tool different from the substitutes that make up this theorem. So, the other thing you need to know is that the Galois theorem is... I did not find it, I am not going to tell you. So it's a string theory, and on top of that, we can say that Serré's superior algebra course was constructed. In the years 1895-96, at the time of the Euler's manual, in France, there are three books that, in the space of two years, have published developments on Euler's theory, which are addressed exactly to the same audience as Serré. That is to say that they address advanced students in their mathematical training, for example students who prepare the navigation, or students who are eloquently written in the intros. The books have in common to stand out from the approach of CERRE even in the fact that they want a new version of El Doriquelwa that takes into account recent developments, and in particular recent developments that we find in Argentina.

10:00 In particular, it is important to integrate into the training of the French students what we have put in the manuals of Weber and Petot. If we look at the authors, these are all three people who are only from the Normal School, who are from the Normal School superior to the same period, who have had the same professors, who are inspired by the same sources, since the sources are explicitly tied to Weber and Petot. A priori, we could say that we are going to have the same type of knowledge. So here too, what is possible is that we can send it to Paris. So I'm just going to present two of them, which are written as Algenic treaties. The third one, for information, has been published. In fact, it is part of Picard's analysis course. An Algenic treaty of anatomy, one chapter. A fairly important development of chaos theory, There is a first book called Lessons on the Algebraic Resolution of Equations which was published by Henry Vogue in 1895. Henry Vogue is someone who did his career in Nancy and who finally published quite a few books in algebra. He made very well-known manuals on mathematics in the age of physicists. He was someone who rather worked in schools of application. This is also someone who has done translations for the Encyclopaedia de Mathématiques de Fein, he has done translations in French, of the French language.

12:30 So if we read the title, obviously what he proposes is not a Gallic theory, it is already a theory of equations. But in there, when we look at the table of contents and the details of the book, the contents of the book, In fact, everything, systematically, everything in this book relates to Galois, especially in the introduction. I did not come here for an introduction that is written by Camus, where he explains. No, it's ours. It's the preface. You should have called it the two prefaces. That's what's funny when you see the... In fact, it was really... These two books should have looked alike because they really had a theoretical component. Penrose did the two preparatory lectures of the two former students who have been here for a year. In the introduction that Penrose gives to the lecture, he explains that what is important here is our way of seeing. When we look at the table of contents, when we look at the denomination of the definitions of the theorems, it is systematically theorems of Gallois, definitions of Gallois, groups of Gallois. In this book, for Ritchie, what is new is that the Gallic theory, contrary to what we had at Serret, is no longer a part of the theory of equations, but it has become a theory of equations. It is presented as a modern theory of equations. They exceed the needs of algebraic resolution. There are things on substitutions, there are things on groups, for which, after all, we don't need to solve equations. So, what does he include in his theory?

15:00 He includes the notions of groups, the notions of injunctions, the notions of resolvents, of domains of rationality. All of this is part of the Galois technique in the same sense, in the same way as the question of resolution. In particular, I will not go into more detail than that, but what is remarkable in it is that there is a beautiful synthesis of what Kronecker and Jordan were doing. It is quite remarkable that in a book, we really see it embedded. A tradition from Jordan and a tradition from Cronenberg. But I will keep a little bit more time to finish with the last book that particularly impressed me. So, the last book contains another exposition of the law, the theory of the law, finished in a book called Introduction to the Study of the Theory of Theons of the Superior Algebra. This book was written by Émile Borel and Jules Dracq in 1895. They are two students of Tendry. It should be noted that they shared the work because Borel took the part of the theory of numbers and Dracq took the upper part. So it's supposed to be a short story. Well, the one who professed Tannery was not normal, but in fact he was largely re-written by Borrell and Braque. So what you also need to know is that when they are re-writing this book, the theory of Galois is in fact one of the fundamental concerns for Braque. This is an application of the Galois theory to differential equations. So, more precisely, while Picard explained that he wanted to import the Galois approach,

17:30 that is, this idea of group and junction, and the correspondence between a group and an equation, The principle of Picard was to import this idea of correspondence between a group and an equation into other types of equations. And Drac, on the contrary, seeks to do something much more general. If we take an article that he wrote two years before his death and that he presents at the Academy of Sciences, it is an article called On an application of the theory of Newton's groups. And he explains in this article that he intends to highlight what he calls, I quote, the fundamental idea that has guided his researches and the generalities of the possible applications of this idea. And then he specifies that this idea allows to present the results in a form as elementary and intuitive as possible. There is something very general in Galois' theory. And precisely, his book of 1895 is an exact transposition of the pedagogical principles according to which he conducts his research. That is to say, he exposes in chapter 6, so it is to expose in the following lectures, the theory of algebraic equations as it was in the pursuit of the research of Gauss and Dabell, built by Galois. In this lecture, I will talk about a theory of Galois, according to which we can reorganize the whole of algebra, the study of symbols defined by equations, the goal of algebra.

20:00 What is remarkable in this is that, for example, if we take the notion of group, the notion of group is part of a notion that redefines for example... There are also a number of algebraic objects in this manual that are presented in mathematical form. There is also a chapter devoted to the polynome and algebraic fields where he defines an algebraic domain of integrity from the notion of adjunction. And then, two methods to calculate algebraic numbers by successive adjunctions or simultaneous adjunctions. And finally, it comes to the notion of group of an equation. Finally, the notion of group is given at the very beginning, while the developments on the algebraic term are given much later. It's quite separate. On the other hand, the chapter on substitution groups, contrary to what was done by Serret or Jordan, is relatively autonomous and has relatively little importance in the book. Drugs are really interested in groups in general and not in particularities such as substitutions. It is a fairly autonomous book. So, if we look at the sources, we see that they are based more heavily on mathematical tradition than on French mathematical tradition.

22:30 To simplify, when we look in particular at the part that is dedicated to substitution, We really have the impression that he is talking about this theory only because there is no other way, because we cannot address the Galois theory without talking about the theory of subsistence, but in the end it is really not the object that interests him. What really interests him is the notion of adjunction, the notion of domain of rationality. This is what makes me say that he was still inspired by the rationality of the world. At the beginning of the 19th century, Braque went on a trip to Germany to study. He had worked with Huxley on all kinds of things, even if we don't find them in the book. By the way, Tadrys is joining us in our production. I will photocopy it. Precisely, the fact that Drac succeeded in bringing to France ideas that until now the French mathematicians had difficulty assimilating, ideas that are, according to him, directly inspired by Kronecker. He explains that I am in the middle, almost there. So, he explains that the mode of exposure of Braille consists of looking at Braille, as well as an anti-positive or negative, etc., as signs or symbols entirely defined by a small number of probabilities set in theory relative to their modes of composition. Something to keep in mind. And what is very amusing is the following. In fact, Henry explains that precisely this way of seeing things This is not in line with what we normally do in France at this time of the year. He explains that there were some fears of having a century of symbols, and that at the beginning it seemed to him that it was not a way of doing that was in line with what we had hoped for.

25:00 There are several passages like that. Where he explains that what Drax does is not in the French habits. To conclude quickly, what I would like to insist on here is that in fact, at the beginning of the 20th century, I have the impression that the Galician theory is an object fixed once and for all. I would refer to what Junrach did. She covered the whole of Algiers. Whereas, for example, in Gott's manual, it is only a modern theory of equations. It should be recognized that there is a set of contents and notions on which most authors agree. A kind of hard core. This is the basis of substitution theory, but without going as far as we have done so far, which is made up of notions of adjunctions and domains of rationality. This is something that appears in France at the end of the 19th century. So we see here that the German and French traditions begin to intermingle at the end of the 19th and beginning of the 20th century. I think that the study trips of other students are not enough. Nevertheless, what is striking is the hierarchy between the objects, the hierarchy between the results, the status of the objects of the results, the order in which things are put, what we consider to be part of the Gallic theory as not being part of it, is absolutely not the object of a consensus. And also, the demonstrative practices on which the theory rests, the way we have to demonstrate things, in particular to prioritize the point of view of group adjunctions, to prioritize the point of view of algebraic calculations,

27:30 is not at all unified at this time. While all the authors tend to claim what Gallois does, there are some who remain very faithful, who precisely complete the work of Gallois, deepen what he did and there are some who give the theorem but who present proofs that have absolutely nothing to do with it and which do not reflect at all on the logic of the theory. All of this to say that at the end of the 9th year of the 18th century, there is still a great freedom of interpretation around a small hard core that we qualify as belonging to Galois. In this context, of course, we can wonder about the influence that these different contributions can have. What you need to know, as I said at the beginning, is that, for Weber's manual, it is an important reference, but if we look in France, in the end, neither one nor the other of the works I have just shown have been re-edited. In any case, I did not find traces, I searched in various catalogs, and I did not find any reissues, neither of Vogt's manual, nor of Braque's manual, which is for her, while the Serret's manual, on the other hand, was published in 1928. This is certain, we find. It leaves us to think that even if there have been many modern, innovative, or re-innovation attempts that have been carried out, even if algebra has greatly evolved, At this time, French students in the beginning of the 20th century worked with Serret's manual, a much more classical and traditional version of the thermal theory. It reminds me of the testimony of Dieudonné, who is very famous, who explains that... He explains that he graduated from a normal school without knowing the notion of mathematics. It is not very surprising because he is very likely to have worked on manuals of a tight type,

30:00 where there are not very perfected or very nationalized. I've seen you, I've seen you. I know you've seen me. I know very well you've seen me. If there are no other questions. But yes, Marie-Josée. Ah, I thought you were going to say something. You don't know this person to ask the questions. Well, let's start with Marie. Yes, it's a small question. Anyway, thank you very much for your exposure. In the same way that you showed that there was no unified presentation, not more in France than in Germany, could you clarify what you said at some point, saying that in Germany we did not have the same way of seeing things as in France? Symmetric and concordant testimonies from the 1870s and the 1880s, In the Treaty of Substitution of Jordan, Jordan began to work before 1870 on the question and published it in Germany. In fact, he explains that it is a great shame that he did not manage to understand what it meant to know. And in parallel to that, So I think it's a so-called Fackmann or Bachmann, something like that, which is an article on Galois' theory in the 1880s, and which explains in a totally asymmetrical way that what Jordan does, well, frankly, he doesn't understand. And it's true that when you look at the lines, you can see very well that in fact he doesn't work at all with what Jordan does.

32:30 He would have liked to be able to do this, but it's not compatible. It's a form of incompatibility. And another little question, how is the concept of body defined? By adjunctions? That's it, not by properties. It's adjunctions and stability of operations apart from adjunctions. It's defined by stability. Yes, stability, yes. Stability, yes. It seems to me that the division between one side of Jordan and the other is a kind of rupture of symmetry, because in Cauchy, for example, there is a kind of symmetry in the subgroups of the other groups, and in the subgroups there is always a portion that is more variable. All of these terms are related to the subject of the group, and when we have a function, we have the structure of the group, etc. So, to connect and connect, we have only one function, which is to be able to place the group in the same order. So, it is clear that there is a difference. It is obvious that there is a difference in principle. There are two completely different ways of looking at mathematics. Yes, these are two completely different ways of taking the same starting point, that's what I was interested in, that they take the same subject, the same starting point, and indeed they develop in two radically different ways, opposites, and what we see quite well in there when we study in detail, is that Jordan relies precisely on... There are a number of values that take a function, etc. While NETO, it's very interesting, there is a lot of incompatibility because NETO addresses the question from the point of view of Kroniker with... It's really... So, a priori, it's the same subject and what is striking is that, a priori too, NETO says that it is used in this day and age. But, concretely... He says that he uses Jordan. Of course, he does not stop quoting Jordan, but in fact, concretely, he does not use it at all.

35:00 That is to say, it is not that he does not use it, but if he has the choice, he will not use it. And on the other hand, he also tends, for example, to take up the theories. He says that he uses Jordan, but he will not demonstrate it as Jordan would have done. He quotes it because it is a reference book and it is difficult to deal with the question without talking about Jordan, but it is not about it that he wants, in fact, that he is not naturally led to use it. Sidiou, it is about these uses of German. It is important that we forget the languages ​​in which these things are written and that we look at the whole of the actors you will talk about. And supposing that we try to establish the resemblances of families, it seems to me that we would put Serres, Jordan, Crandon, Crandon in a family, in any case closer, that we would put Debeckin, Debeckin, so it seems to me if you want that... At a certain point, taking Kronenberg to say it in German, there are certainly actors at these efforts who say that this is the only way to do it, but there are other things in German, so I would prefer that you take back all the pictures that you have introduced. To say that there are different types of traditions in the language and in fact that these traditions do not necessarily exist in the language but in terms of ways of doing things. It seems to me that in what you have shown there are really things that circulate and that make a family. Yes, it's true that it's part of the languages, but anyway, it's explicitly the French manuals, very explicitly, the manuals of Aton, it's the goal displayed.

37:30 The goal of the Vogue manual, for example, is to do something accessible to French students, so in French it is always easier, but which is clearly inspired by what we do in Germany. So there is a will... I thought it was not necessarily Vogue, but I understand that there is not necessarily the aspect of editing. But both, in fact, even Drac, in fact, Tannery when he says... Did you say that Drac was inspired by a tradition? Yes, Drac. Yes, Focht. I just had to understand because you said it was more the chronicle side. No, it's more the chronicle. Yes, both in fact. As much Drac as Focht is inspired more by the chronicle side than by the equine side. In any case, that's what they say. After that, it can be discussed. It seems to me that you have introduced yourself tools and distractions that would lead you to use rather certain types of traditions rather than German since, as you said yourself, you opposed Göttingen, and so, in fact, I think it would be more consistent with what you said to try to continue to try the words, it seems to me. In any case, in France, what changes in the French manuals is that there is also this importation from the Serres that we do not necessarily find in the German manuals. There is a mix of different cultures operating in this area. The German manuals are clearly not, the French manuals, by the way, are clearly not in a continuity when we take up the system. So, the other aspect that struck me, if you will, is that at the beginning, in the first texts that you mentioned, we are not talking about theoretically, and all of a sudden it appears, and in fact, I quickly looked at what you showed, and I had the impression that you were talking about theoretically. No, in fact, theoretically, it appears a little bit in...

40:00 We start to see the expression in the 1860s, 1870s, Galois theory, in fact we talk about it in Jordan. Jordan is the first to say that Galois made a theory. What struck me in the field of mathematics in relation to science, if you will, is that we have theory of rationality, theory of Galois, so we have the impression that the structure of mathematics takes theory. And in fact, today, we would say theoretical theory, but I do not understand that we would say rationality theory. So, in fact, we have the impression that there is a moment of structure of mathematics in theory that will not have the same devenir. And I wondered if, well, obviously, it's a question that is not too broad, but if someone can give me the answer, if you want, what... The correlation between the way it perceives its phenomena and the fact that there is a theory to qualify it. In any case, from this period on, before these different... Before, indeed, when we talk about Galois theory... It's really not something that is related to the structural gene or anything like that where we don't see everything that we have been able to add now from a general point of view. When we talk about Gallic theory before, it is more a reference to a way of solving something that is related to the theory of equations. It is fundamentally related to the theory of equations in a very complicated way. We often find, for example, the theory of equations according to Galois. This is an expression that we hear very often. So it's really like that. Galois theory in the encompassing sense is something that is posterior. I stopped my studies at the beginning of the 20th century. I would like to see if it is not related to the reflection of Galois theory.

42:30 We should have talked about it. I would be surprised. I wouldn't be surprised if the name of the term was a bit overwhelming in the end. But if you want, it deserves an interpretation. The emergence of this new theory and its different meanings. In fact, everyone calls it Moscow, everyone calls it Thoreau. Everyone doesn't put the same thing behind the words of Gallois and it's probably also related to what we hear by theory. So I think that's also what it would be interesting to hear. Because I think it would be of general interest on... What we can say by... what it means by theory. I have two questions to listen to. And did you look at the side of the posterity of Gallois' works in England? I watched Keyer and I went to see Mr. Kirkman. I went to see what they called the... I forgot the name of the tactic. It exists, yes. It exists, yes. Kirkman is the one who is the most important. He is the most important. He is the one... Well, in fact, it's... When I was talking earlier about the 1860 Maths Grand Prix in France, there were two Frenchmen, it was Jordan and Mathieu, and there was... Oh, no, no, no. And so I looked at what he had, and I saw a chapter, a mathematician, yes, British, and who developed an extremely original point of view on the notion of group in the 1860s, it's extremely, it's egotistical. It's in your thesis.

45:00 When did the notion of group of volumes appear? The conceptualization of particular properties of substitution groups in this language is very important. In fact, I wanted to come back to something quite different in your exhibition, which is the opposition that you have made at the moment between manual and written. Because, if you want, what you called manual, you said it's something that follows the course a lot more, it's divided into lessons and, in any case, it's a good example of something like that that evolves towards a more integrated one, and it seems to me that As much as the manuals of the treatise, they are texts that are produced in relation to the teaching activity, but perhaps these texts do not have the same meaning. There are many links to the teaching activity that are not used in the same way by the teaching activity. What I wanted to say about Humanities of Ceres is that the first edition is almost someone who took the course in notes and published it. It's organized like that. In the second and third editions of 1966, which moved a little afterwards, there is still a work of distancing between the printed work and what we could have seen in a group. It is not organized. For example, the 66 manual is not necessarily linear. We can very well imagine that the 66th manual can be taken from one place to another, and it is more a tool of work than something that comes directly from a word. If you want, it seems to me that in your story, well, of course, as an anthropologist you have a choice, but it seems to me that in your story there is something interesting about this report,

47:30 So there are really texts that stick together a lot and in both cases, one time it's with Weber and the other time it's with Serre-Lebel, there is an evolution towards the form of the treatise, but I think, first of all, I think that this material Serre-Lebel is very interesting to analyze the passage, to analyze what is the type of treatise. All of this is linked to the activity of teaching, that is to say the necessary digestion that the treaty produces and which allows teaching. On the other hand, there is a problem that does not fit completely. The problem is that the 66th edition added the manual, the first manual, if you want, the manual of 49, it's a little thing. There is a lot of content that has been added to the 66th manual. The 66th manual is something that is really wanted, a reference book with two volumes, it is much more complete, there is really, the 66th manual is a bit of everything you need to know about the superior algebra. Yes, I would like to ask if you want, is there a prerequisite? It is a form of activity that is required by teaching.