 I realized I allowed them to publish the title in French, which was fine with me, but I don't want to mislead anyone. Yeah, talk about what happened with Gaoua theory in a good in school. This was very important to Arten. We have this quote since my mathematical youth under the spell of the classical theory of Gaoua, returned to it again and again to prove its fundamental theorems. We have Zassenhaus saying, I was a witness how Arten gradually developed his best known simplification. Of course, this is not all the work Arten did, but it's the work on Gaoua theory and it's a simplification. His proof of the main theorem of Gaoua theory, the situation in the 30s was determined by the existence of an already developed algebraic theory initiated by one of the most fiery spirits that ever invented mathematics, the spirit of Gaoua. This is a lovely thing to say about Gaoua and it's a lovely thing that Zassenhaus is saying, but I have to complain a little bit about what Kiernan makes of it. Kiernan in his history of Gaoua theory sort of suggests yes, this is what was happening in German algebra in the 30s was Gaoua. Well, no, I mean really the whole theory of algebras was going on and absorbing this. Kiernan, Arten took a revolutionary new look at the theory, took up the concept stated implicitly by Gaoua and announced unheard by Dedekind and Weber, unheard they announced this. The theory is concerned with the relation between field extensions and their groups of automorphisms. Weber certainly, in his client, in their book, primarily two great general concepts lead to a mastery of modern algebra, the concepts of group and field, that's what it's all about for him and it is true that no general account of Gaoua theory between Weber and Arten emphasized this. Kiernan can correctly say, I find no introductory treatments of Gaoua theory after Weber and before Arten that take this viewpoint. But the progress was not only in the general accounts of Gaoua theory. Kiernan would have it that in the early decades of the 20th century, German mathematicians such as Emma Netter began to examine and detail fields in their generalizations. Well, yeah, this is a drastic understatement. But besides a misspelling of the name. So the question is what did Arten do for Gaoua theory? A lot of things. He generalizes Dedekind's fundamental theorem of Gaoua theory from algebraic subfields of the complex numbers to arbitrary fields. This is typical of what goes on, which especially involves considerations of separability because you now can have finite characteristic. He eliminated primitive elements from the proof. This is the single thing most often said about Arten's improvement. And eliminating primitive elements was important to him. He really wanted to get them out of there and it's a nice achievement. But it must be said that while everyone today uses Arten's statement of fundamental theorem, many still prove it using primitive elements. We had the three strategies for proof mentioned yesterday and it's not unusual to use primitive elements. Van der Waarden in a book which dedicated to Arten on the first page and Arten and Nutter together uses primitive elements to prove the fundamental theorem. What I want to say, the key thing that Arten did was he made Gaoua theory into modern algebra. He brought it into this framework and well thus made modern algebra possible, the book. He built it on degree of polynomials and dimension rather than on calculations. Of course it rests on calculations. It's a theory about algebraic calculations but you don't need to spend all your time doing algebraic calculations to pursue it. Besides understating the substance of Arten's work, Kearnan misses its place in the future of Gaoua theory, the way it was gonna move on in Arten's own hands into class field theory and cohomology where it was going with Arten. Nutter meanwhile had thoroughly absorbed and advanced Dedekind Weber on fields and groups to say that they were unheard. Well, not only is that to neglect Nutter but Nutter points out that they were unheard. Nutter is the first one to say that people have not been following Dedekind as well as they should. So that Kearnan is not only neglecting her, he's following her footsteps unaware of it. I think that the single most emblematic point of this is this really beautiful trivial result that any field K acted on by any finite group is Gaoua over the fixed field. And this is now, we're not talking about the rational numbers, we're not talking about polynomials, any field, any finite group is Gaoua over the fixed field. And Nutter's going to use this. She's going to use this in connection with her own really, really fantastically innovative work on group invariance. She's going to apply this plus her own invariant theory to the inverse Gaoua problem. This is I think mentioned in 1913 and published in 1918. If you let a finite group, so the point is we're going to take a finite group, we want to know if it's a Gaoua group over the rationals. And one approach here, take that, let it permute the field of rational polynomials when in as many variables as there are elements of the group, you just let it permute the elements of the group. Nutter has shown that the fixed field is finitely generated over the rationals. This is a new result in invariant theory. And she's shown that if the generating set can be shrunk to the size of the group, then it's a polynomial ring over the rationals. Whereas the whole field is Gaoua over that polynomial ring. And so this is exactly the situation of the Hilbert irreducibility theorem, which was designed for the inverse Gaoua problem. She's now shown that you can apply Hilbert irreducibility whenever you have that the invariant field has a generating set the size of the group. That whenever this happens, Hilbert irreducibility shows that G is a Gaoua group over the rationals. In fact, it shows much more than that. It gives you a parameterized family of a sense of all except with a few exceptions polynomials that have that as Gaoua group. This has become now Nutter's problem. And it's still a standard approach in the inverse Gaoua theory. It does not give a sufficient condition for a group to be a Gaoua group. Well, it gives a sufficient condition, not a necessary and sufficient condition so that the problem of which finite groups or Gaoua groups remains open, but this is certainly an important way to look at it. And it depends right here. I mean, on this fact, which she knows very generally, as well as her very sophisticated recent work on invariance of actions, Nutter has entirely absorbed the data conveyor atmosphere, and Arten is, of course, learning it from Nutter. I didn't put it in my notes, but Natasha Arten explains Arten's method for learning from Nutter. It's a lovely story. What you do to learn things from Emmy Nutter is you go out walking with her and you ask her a question and you walk and walk and walk and walk while she answers it. And after 20 or 30 minutes, you say, but Emmy, I didn't quite understand. Can you tell me again? And you walk and walk and walk and walk and walk while she answers it. And you repeat as needed until about the third or fourth time when she's become somewhat tired and slowed down a little bit in her explanation. And then you can understand it. What I wanna say is most important for understanding Arten and the future of Gaoua theory, Arten's role in the future of Gaoua theory, is Nutter's work on algebra up to isomorphism, but really not the general logical sense of up to isomorphism that we find with model theorists. I love model theory, but not just that. Certainly not what we find with structuralists. Nutter has a very particular conception of how to do algebra, which in hindsight we call up to isomorphism, what she called purely set theoretic and independent of any operations. What she means by this is that you're gonna look at inclusions, you're gonna look at modules and their submodules. You're gonna look at induced maps between quotients. You're gonna look at ideals of rings, which are of course submodules of the ring as a module over itself. You're gonna let these inclusions of submodules and induced maps between their quotients replace the elements and equations. She's specific, this is without elements and equations. Relying very much on her homomorphism and isomorphism theorems. What today we do with exact sequences. This is approaching homological algebra and that framework. Except that Arten is working over fields in his work on Gaoua theory, which is not all the work he's going to do, even relevant to that. Since he's working over fields, it all turns into dimensions of vector spaces. We simply have counting arguments that we'll do for homomorphism and isomorphism theorems. Arten gives this idea of a character of a group as a group homomorphism from that group into the group of multiplicative units of any field. He shows that any list of characters is linearly independent. This is an appalling result just take any list of them. They'll be linearly independent. Now, this is because they don't form a vector space, right? Obviously, you can't have a vector space where every subset is linearly independent, a non-zero one. But because they don't form a vector space, they're all linearly independent from each other. And of course, so this then puts lower bounds on dimensions of certain vector spaces. I could go back to that statement. So that over and over again, he'll take some collection of polynomials of interest or some collection of points in a field extension that's of interest, he'll represent them in terms of some characters of some group and he will then establish a lower bound on how many of them must be independent on the dimension of the vector space they form by the independence of the characters. But lower bounds on some, he does not use the word space. I realized that when it just now working on the notes. I keep calling it space, he doesn't call it space. He just talks about linear independence. Lower bounds on the dimensions of some spaces give upper bounds on the dimensions of other spaces because if one of your spaces was too large, if it had too many dimensions, it would give you freedom in which to work to provide solutions of certain linear equations which would impose too sharp a bound on some other space where you've already got a lower bound. So his independence result, which is prima facie about lower bounds also gives upper bounds. And so what he proves is not equations between elements but equations between orders of groups, indices of those groups, that is the orders of the quotient groups, indices of normal subgroups and dimensions of spaces. He turns it all into counting arguments on these things and thus shows that one field extension equals another because one is included in the other, they have the same dimension over at the fixed field, things like this. But Netter, along with notably with, oh, Brower type over there and Hassa was already using what she calls cross products and representation modules for class field theory. This approach by way of algebras, hyper complex systems to represent, really finally to represent Galois actions and apply this in class field theory. Most famously Netter's Hilbert's Theorem 90. I tried to state Hilbert's Theorem 90 and there may be people who can do this. To me, Hilbert's Theorem 90 is more puzzling when you first hear it than not. You can present it. And you can show that, for example, it gives you a derivation of all the Pythagorean triples but that's not really a helpful view of what it does. Units of algebraic number fields then have to come as quotients of numbers and their conjugates and that turns out to give the Pythagorean triples but that's actually more of a bad thing to tell you than a good thing because that's not really what the theorem is about. What I can tell you is it shows that certain first cohomology groups are trivial but that's also not helpful. And she was consciously bringing, she's doing this to bring Galois theory into her framework of rings, ideals and modules using what she calls the purely set theoretic foundations for them, or purely set theoretic foundations. I wanna look back a little bit at where this comes from and then look forward again to understand the 30s we have to appreciate how estate all is shown by Dedekind. I've said before and I will continue to say Emmy Netter's statement that this is all already in Dedekind is the most profound statement ever made about 20th century mathematics. It really remains to be understood. It needs to be understood. This is all already in Dedekind and nobody would have known that if Netter hadn't pointed it out. Not only, I mean nobody would have found it there without her. This is a fantastic collaboration of mathematicians, Netter and Dedekind who apparently never met. There's a little bit of room to think maybe we just lost the evidence because their careers overlapped but if there's no evidence, they probably never met and yet Netter works so directly with his ideas and so consciously working with his ideas. Dedekind makes the crucial observation that algebraic independence of a quantity A over some field is linear independence of its powers. Algebraic independence which looks like a very hard property is linear independence which looks like a very easy property. They're the same property. Of course, logicians will notice that this is linear independence of an infinite series. So there's room for it to be harder. Dedekind proves that for any system of n permutations of a field, automorphisms of a field, infinitely many numbers in that field have n distinct images under those permutations. Artin would later make this a result on characters. He'll say a permutation after all is a character. It's a group homomorphism from the multiplicative group of the field to itself. But Dedekind doesn't define, as far as I know doesn't define characters as including infinite groups. It's only characters for finite a billion groups. Artin extends it to all groups so the above becomes a case of independence of characters. Independence of characters is a stronger result in this in a way, except he doesn't mention point wise independence which is also not a hard consequence. So that here we have, I mean it's a little bit interesting to watch this develop, but it's not a crucial change except the generality, Artin puts it so much more generally. Dedekind, what's the real difference to watch between Dedekind and Artin is that Dedekind is inventing the idea of linear independence right before our eyes, not for the first time, not for the last time. This is an idea that will have to be invented many times including by Dedekind, but you watch him. In these paragraphs 164 and 165 of one of his appendices to Dirichlet, he changes his terminology in the course of the paragraphs. He reformulates results in the course of the paragraphs. It's a really beautiful process to watch, but it's a laborious process. It's a difficult. Casts around for the right motivations, the right definitions, the right terms to express them, the right theorems. For Artin, on the other hand, this material is perfectly mastered. Artin takes this up and uses it beautifully to simplify the theory whereas Dedekind is there inventing it. I like something Peter Fried said that we too often try to find the first inventor of some mathematical concept and we too often strain ourselves trying to sort of read inventions into things. He says, what's important is to find the last inventor of the concept. The person who invented it so well that it never needed to be invented again. This person also turns out not to exist, but it's a good perspective to have. And here we're watching one of the inventions of linear algebra. It's a beautiful thing to see. Looking ahead, Artin, especially with Tate, is gonna take this into class field theory and Galois co-homology, which is where Netter is also taking it, and Brouwer and Haase, and this will work. It will go ahead with that. Ser and Grotendiek will restore the link with monogamy and Riemann surfaces with their isotrivial and etal covers, co-homology and fundamental groups. This is gonna go back to that. At this point now we're thoroughly using, well, the descendants of Netter's homomorphism and isomorphism theorems. We're now using homological algebra. We're taking this back in more in the direction of Netter's own work. That's where my slides end. I would say a couple of things that come out of this that stood out to me. I have not researched Dedekind's development of Galois theory, and I liked it when Caroline Erhardt mentioned Dedekind's lectures on Galois theory. It would be interesting to go more into that, and I haven't. And a really crying gap here that I don't feel in a position to make up any time soon is the connection between algebra and differential invariance, which we're just talking about out here. Netter probably grew up very aware of Sophus Lee. Sophus Lee was a local hero in Erlangen. Netter's father's a professor at Erlangen, her, I don't know, godfather, and Paul Gordon, who would be her dissertation advisor. These people took Sophus Lee very seriously as a local hero. She will have heard of him from the start. Just about the time that she's doing this work on the inverse Galois problem, she's also doing her work on differential invariance and on conservation laws and physics, which you look at it, she describes it as a continuation of the work of Lee. As I've said before, Emmy Netter is not a modest person. We misunderstand her when we think she's being modest. What she is is a person who's actively engaged with her predecessors. So she's not underrating herself by saying I merely followed ideas of Sophus Lee. She's telling you how she experienced this. She was in conversation with Sophus Lee, another person she never met in face-to-face. She was absorbing these ideas, and this is what led her to that work. But the connection is gonna be very deep, and I don't think easily found. So this is, in a sense, not even Galois theorian get again. This is Galois theory deep in Emmy Netter's mind. And I haven't really been able to draw that out. So that's what I have to. There's a paper by Steinitz on field extensions. I really would have thought that that would have influenced some of the later work. I remember reading some of that when I was a student or just after that, and they're just beautiful. Mm-hmm, mm-hmm. Netter is very involved with what went on before in field extensions, and of course, Artyn also. I don't remember Netter citing Steinitz so often, but she had absorbed all that material. This was definitely her approach to a problem was to read everything about it. So I suspect you're right, there's a lot to know about that, but I didn't look into it. I'm sure you're right though. A very small question. Could you explain why Artyn wanted to get rid of the primitive element? We have Zassenhaus's description of this and Artyn's description of it. The published descriptions make it pretty much an aesthetic point that he believes that Galois theory is about field extensions, not about solutions to polynomials, although actually that doesn't really bear on the question because if you see it as a theory of polynomials, normally the polynomial you're interested in does not have primitive elements as roots, so that you have to pass to a resolvent to get that. If you look at it as field extensions, then of course the problem with the primitive element is it's kind of arbitrary. Virtually all elements are primitive and how did you pick this one to pay attention to? So I think his feeling was that sort of aesthetic one that we don't like to make the fundamental theorem rest on an arbitrarily chosen primitive element. We would rather make it rest and then especially when you see what the alternative is that instead of resting on choice of a primitive element you can make it rest on counting the dimensions of spaces in the field extension. You have a real pretty alternative and the fact that especially since that invariant alternative was so nice, he wanted an invariant account. I don't think there's a mathematical motive here. I don't think there's a feeling that we'll get important new theorems by getting rid of the primitive element. And of course primitive elements are not eliminated from the theory at all. They remain present for the things that they're used for. I think he wanted it out of the fundamental theorem really as he said, let me get back to the first slide there, because he was under the spell of the classical theory, he thought it would be nicer not to have it rest on an arbitrary choice. And Fonder Varden does not eliminate it himself. The very first person dedicating this book to Artin and Nutter does not eliminate the primitive element from proving the fundamental theorem. Other questions? Concerning your last remarks on her work on conservation laws, I think it would be very interesting to understand how in her mind it was related especially to her recent article on the inverse Galois problem. So one might wonder why didn't she invent differential Galois theory at that time? I would be tempted to say that her 1918 paper was so powerful because she looked only at variational problems and not differential equations in general. We don't know that she pursued any or maybe you know whether she pursued any work on differential equations and symmetries later in her career. So this is my question. No, I don't think she did. In addition to her truly important great paper on this she's got a note. It might have been in a second edition of the Encyclopédie on algebraic and differential invariance. She does see it as a problem in invariant theory and so part of the answer to the question would be that she had the wrong approach in variant theory didn't turn out to be the right way to look at that. I think the real reason that she didn't pursue it is given by Alexandrov though. I think Alexandrov is exactly right when he says that she decided that her true calling was to create a new algebra. It's not that she couldn't have pursued the invariant theorems, their beautiful theorems but she had been given them as an assignment. I would say in some sense that's her last student work. Hilbert and Klein had said here's a problem you should do this and she did that problem with real genius. But after that she picked her own problems and she picked them in commutative algebra and on the way to class field theory. But I think yes she, to me when I read the invariant theory paper it also shows her method that she went out and learned everything on this subject and then she pulls it together really fantastically but she does in her own comments on it some years later which were probably, she would hold on to an idea for years before she actually released the publication. So probably from the start she had seen this as a kind of work on differential invariance and that perspective didn't go as far perhaps. I think, I will mention something that seems to me important about that work that I don't understand at all. I think that Emmy Nutter was really not a visually geometrical thinker. I believe that even in her paper on invariant problems she sees formulas she does not see curves and surfaces. Now she sees formulas and they're beautiful formulas but and she was very fond of the fact that this was formal calculus of variations. That's and it's also a beautiful idea but I think she doesn't see it as symmetries in the visually geometrical sense that say Richard Feynman does when he later writes about her work on it. I think she sees it as formulas and that puts a context on the work. Maybe the last question. Just a remark about this. As you know when people speak about the possible geometric content of Galois everybody gets hopelessly vague. So obviously he's very vague. Picard is vaguer and so when you come to Nutter if there is something geometric in Galois it's unlikely that it would involve the Hamiltonian formalism. But it would seem to me that the conservation laws have very little to do with Galois. If there is something it's probably what we will hear about differential Galois theory and so on but I don't think I mean Nutter did that, did she? No, I have no reason to believe that she read Galois. She read Sophus Lee and she read Sophus Lee about Galois but this is precisely Sophus Lee about Galois. And also I would call it geometrical if it's just not visual geometry. She has an analytic understanding of geometry, utterly analytic to the exclusion of visual but I would still call it geometrical. This is a geometric question but I will ask you because I'm sure it will be covered later. It is when people start understanding that uniformization, ramification geometrically is the same as what happens in number fields but I'm sure we'll hear about this. Premature. So thank you very much. Thank you.