Hello, Everest, Gaulois, jimm and revolutionary, but it says on a commemorative French stamp made in honour of one of the most enigmatic and unusual mathematicians in history, he lived a very short life in the first half of the 19th century, dying when he was just 20 years old. But during that brief span, he produced work that, after his death, would go on to revolutionise mathematics in the years to come.
Today, we will not only discuss the extraordinary details of his remarkable life, but also give a flavour of the kind of mathematics that his ideas gave rise to. With me to discuss Everest I working his life are Chris Nichols, a third year student in maths from Baylon College, and Benjamin Green, a final year student and lecturer at Baylor College.
Thank you very much for joining me. I think if I look at the plan that we've made together correctly, we're going to alternate between some vague details of his mathematics, but also the details of his life. Chris, when was Gaulois born? Can you tell us a bit about his his upbringing and where he came to mathematics? Sure. I can try and say yes. A, I was born in 1911 to a relatively well-off family, but they didn't have any history of mathematical ability, in fact.
So it was quite surprising when he developed this philosophy taught by his mother until about the age of 12 and his father was eventually to become the mayor of this town when Gaulois does eventually go to school. Is he a successful people or would his teachers think of him as teachers? Aren't actually that impressed sometimes, although they're their comments that are not always that consistent. For example, one teacher actually describes him as singular, bizarre, original and closed.
But in fact, it's exactly this originality that we remember him for. So, yes, I think the word it's a wonderful word, singular, because if you read a lot of, say, Arthur Conan Doyle, Sherlock Holmes books, he's singular to me like an odd occurrence, which is certainly never used in the English language now. So whether being singular is good for a people is certainly certainly up for debate, I guess.
But it's funny. I would say that all four of those attitudes describe Gower's work just to achieve his teacher really huge perspicacity. And whether he meant it or not, he discovers mathematics when he was about 14, 15. And this has a huge effect on him. And that's something that his teacher said about his mathematical ability at that time. Right. Right. So his teacher at the time when he actually enrolled in his first class said that it is the passion for mathematics which dominates.
And I think it would be best for him if his parents had allowed him to study nothing but this. He's wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments. So it sounds to there that he said if you found his message, he sent it, went for it. Yeah. And I think at this time, he's reading a lot of mathematics books that are that are out there. But a lot of the things that he's interested in actually just aren't very well developed at this time.
So what he eventually becomes famous for is his work in algebra. But a lot of the textbooks that really aren't so well developed as a modern student would seem to me, if you read a modern textbook on algebra, it would be very, very different to what you would get. If you read a book at that time, then what you say was the mathematical culture of the time, the mathematical inclinations of people in Paris.
Well, I think a lot of mathematics then was developed to try and solve quite practical questions. So like people like Watson or Fourier, they have famous mathematical results associated with them, but which were often used or developed to try and solve practical mathematical problems and also practical real world problems, which had a mathematical angle.
And what Ghawar himself became famous for, although not during his lifetime, many years after his death, was for much more abstract mathematics. And that is essentially taking a question which may be quite natural and trying to look at it far removed from the original problem.
And as Chris mentioned, a kind of algebra as a modern mathematics student would learn about it would have been very, very different from the way the GAO or even people, you know, say, 10, 20 years after his death would understood algebra wasn't properly developed, maybe until the late eighteen hundreds in the manner that most students now would consider it.
Exactly. If I recall correctly in was it eighty eight when Fourier introduced the phrase transform as we now know it, which is a major method of mathematical analysis, it was to to help solve a huge equation and understand heat propagation in body. So it was, as you say, a very practical question that what is now a purely mathematical method, but it was motivated by very practical problem.
Let us now do our first little stint of some of the actual mathematical content of Galileo's work, then mentioned to us that Gauloise work within algebra in particular is involved with solving equations and a particular kind of equations called. Polynomial equations, and that's a very long word, but Chris, that's not such a scary concept. It's very similar to what many of our listeners would have studied at school. Absolutely.
Yeah. So, I mean, examples of polynomials are things like, well, one, two, but also things like X, X squared, X squared plus one, this kind of thing. And where X is some unknown. Exactly. So actually some unknown quantity. And I guess you can think of it as it's a formula that you can substitute some value into if you wanted to. So you could look at something like X squared plus one and you can say what that take effect is one you can access to.
And the thing that Gauloise really interested in is what kind of solutions there are to this thing equalling zero, these solutions called the roots of the polynomial. So I think at school, people may be familiar and maybe heated this quadratic equations. This is where you have A squared. So X squared plus three, X plus one equals zero. Exactly. But but you're saying but the interesting question is when you have maybe X cubed extra before.
Exactly. So. So as as maybe you remember from school, I mean, not you, because I'm sure you do remember that the quadratic formula that gives you a way to solve any equation of the form X squared plus X plus B, where A and B is some number. And the idea is that suppose you take some equation like X squared plus two equals one and you want to know what values of X can give a solution to this. So what values of X you can put in that'll will give X workers to express one equal to zero?
And there's even a formula that tells you exactly when you can do this. So you can consider that case completely understood. But when you go to something like a cubic equation, which is something category three, maybe cube plus two X squared plus X plus one, then it gets a little bit harder. But people can still come up with some formula to solve this. Then when was this Kubic formula discovered? This was done by Italian mathematicians and the 1400's, so goodnews method is the standard method.
Now, at least for solving a Kubic equation. I believe that was actually published by code, developed by him basically gives a false formula very similar to the quadratic equation formula that some of our listeners might remember,
which is just slightly more complicated. So the quadratic equation formula, if you have a quadratic X squared plus B plus C, then your formula for it, the solutions minus B plus or minus the square root of this grade minus four is the all of the two I still remember from school mathematics. Exactly. Because it depends on A, B and C and similarly for a cubic formula it will depend on the different coefficients which appear in the cubic format.
So of course there are those four of those and the quadratic case, there are three of those. And fairly soon after mathematicians have developed way to solve the cubic formula, they also came up with a way to solve a degree for polynomials.
So again, you got a slightly more complicated expression now, depending on the five coefficients which appear in a general degree form four a.m. and that and the thing about these equations is, you know, once mathematicians had found a degree for a degree, for one, it was a natural question to ask. Well, if you give me a degree five polynomial, is there a similar formula for this and so on. So a formula just involving the coefficients, I guess there are six coefficients now is a X to the five.
Must be X before the formula, just involving those coefficients. That will give me a solution to this equation in zero. The exact statement about what we mean by a formula is slightly complicated. But you are calling the quadratic formula. Basically you had to add coefficients together, divide by them and importantly take a square root of them.
Now, similarly, in the cubic formula, there are additions, division, multiplication going on and there are square and cube roots and similarly the degree four equation for groups. So basically what they were kind of hoping for was in the degree five case, and if you kept going up that you'd be able to just get a formula involving basic addition, subtraction, multiplication and division and also taking root. So square root, key root for fruits and so on.
And that's now. Exactly. And that and that would be how you could state all the solutions. That's what they were hoping for. OK, so we have from about the middle of the 15th, hundreds this very natural conjecture, I guess, that for any 50 degree polynomial, there is some formula that takes the coefficients and gives you a root. Chris, so Gallo worked in this question and this is, you know, by the early 19th century. So it's been unsolved for a very long time.
And a few years before Gaulois, there was a mathematician called Abel, a Norwegian mathematician, and we can let the cat out of the bag. Now, what did Abel manage to prove to Abel actually proved that there's no formula for the general degree five polynomial in the form that Ben suggested. So it may be that for a particular degree, five four, you can find the roots. For example, if I take the polynomial X to the five minus one equals zero, then certainly one root is X equals one.
One of the five. That's about a one 1.0. But the idea is that if you just give a general polynomial of the form X to the five plus base of the four and so on, then there's no formula just involving roots of some degree and coefficients in this polynomial. Which is amazing because, you know, you have this natural idea where you say, OK, well, I can solve every two problems.
We can eventually solve degree three polynomials and then degree four. And it really looks like this pattern should continue. I mean, there should be some way of rearranging operations, taking roots to get the solution. But this amazing result that is not possible is extraordinary and very, very surprising. I think there's a reason why it took so long for mathematicians to discover this is who would expect that this would be the case.
Let us come back to Galois and his life. So we've left him at the end of school. Then he tries to get into the Ecole Polytechnique, which is the premier school in Paris at the time. How does that go? Well, it doesn't go very well for him. He was rejected from the whole technique, which was something that he was obviously very disappointed about. He, as we heard kind of earlier, he developed a real passion for mathematics.
And the technique was founded by a mathematician in the late seventeen hundreds. So it was certainly the pre-eminent place in Paris and indeed one of the best places in the world to be studying mathematics, because in France there was quite a tradition of good mathematics around. And therefore, yeah. So certainly for for Gaulois it was incredibly disappointing to have been rejected from the school.
So he instead had to go to. Ecole Normal, or I think it was called something slightly different at the time, but that's what it's known as now, which I think you mentioned was a teacher training. Yeah, it was the place where if you were going to go on to become some kind of secondary school teacher, that was where you did your academic training. And in fact, it was housed in some kind of annexe of the building where Gaulois secondary school was.
Yeah. So it was very much, you know, still confined within that same place. So he was very happy to be there. But he tried a couple of times. They're called Polytechnique. And yeah, there was a rather sad that happened in his life just before the second time. Yeah. His father committed suicide in 1829. I think a few weeks after the death of his father.
He actually tried again and was rejected again. And I think also the the is the suicide of his father, I think was related to we mentioned that his father was a matter of his local town and it was related to the various political upheavals that were going on in France at the time.
So that perhaps this was something that, as we're going to talk about in that Gowa was very involved in the or certainly wanted to be very involved in the politics to remove the monarchy or anti monarchy that was going around in France at the time. And therefore, this might have pushed him even closer to being very tied into a political ideology. I mean, revolutions to a penny in France in the 19th century.
But so this July 1830 revolution of the current monarch, at that time, Charles the 10th gets chased out of Paris and no monarchy is reinstated. But it is a time of great upheaval in parts of around this time. I mean, beloved light over lots of details. In early 1831, Gaulois submits a paper to the academy for submission to a prise. Chris, can you tell us a bit about what was in this paper and its history, its story?
Absolutely. So so this paper is actually about the solubility of various equations in the way that we've discussed before, using brute sense and functions of the coefficients. But not many people actually liked this paper because the main statement is that.
If you have a polynomial of prime degree, so, for example, it starts with the five and then it's about two things, or maybe it starts on the seven side of things, then it's solvable in this way, which we can refer to from now on by radicals, sold by radicals means solvable by a formula of the kind that Ben discussed earlier. Exactly. It's solvable in this way. If and only if, whenever I take one of the routes, I can always express it as some function of two other routes.
So, for example, suppose one of my routes, suppose I have a quick one of them and I call my routes Alpha, Beta and Gamma. Then the condition is that I need to be able to express Gamma as something like alpha squared plus beta, all of alpha. So it's some function like this intented, often beta. Technically, I guess I would call it a rational function in beta. But this is peculiar because we talked about trying to understand in terms of the coefficients of the polynomial, what the roots are.
But Gallas rather perversely, he's saying, well, you can understand the roots if and only if the roots have this property. Exactly. So this is not a very checkable condition because at the end of the day, what you're trying to understand is the roots. And to understand this thing, you need to first find the roots. Ibrahim Apriori, he submitted this paper for a prise at the academy. And what happened? What actually happened? I think the paper was lost.
So Fourier, who is I think briefly mentioned before, was a very famous mathematician who was in charge of collecting these entries for the PRISE, and he was involved in deciding he received it actually died at the time. And it's not quite known whether the fact that Galois manuscript was lost was connected to the fact that obviously there was this discontinuity in the procedure because the person in charge of the PRISE died during the decision making process.
But this manuscript that Gaulois submitted was lost and in fact, the PRISE was given instead to Arbel and Jakob, which is in some sense quite a funny coincidence, given the Arbel, in fact, also did work very similar to ours. So this was, in fact, not for everyone to go on. So yes, those Zaghawa and this was but this work that he got the prise for was in fact not this work on solving certain polynomials.
It's also perhaps worth saying that Arbel himself had in fact just died in 18 and 29 at the age of 26. So there was quite a collection of young mathematicians dying prematurely in Europe at the time. There is something about trying to solve equations by radicals leads to early death, it seems.
Yes, this is the inference. Okay, before we start to return to some of the methods that were used by setting this up, how it is Ghale was different from Able's and why are we making sure about Gaulois rather than able? How did Gaulois make a more fundamental contribution to Galileo's contribution?
I suppose wasn't really realised by the general medical community until a few years after his death, but it really set up the path for modern algebra, the first to actually make the link explicitly between whether a polynomial equation is solvable, by which I mean whether we can write down all the all the roots in terms of these radicals and. And permutations of the routes by which I mean take the routes and consider different ways in which you can rearrange them.
So if I had the routes, Alpha, Beta and Gamma, then I could equally well consider the routes gamma, beta and alpha. So it's just different ways of arranging the routes. And Gallimard, the link between. The ways in which these routes can be rearranged and whether or not the equation is actually solvable in terms of radicals, there is like a Ruffini also read this kind of massive book on this subject, which is centred around, you know, read.
But it's this explicit study of these shuffling of the routes and the routes first. I mean, because it's under the surface in the previous work and enables what is first time is really explicitly etched into the surface in Kalamazoo. And the reason this is interesting is because this paves the way for much of modern algebra in terms of studying things abstractly and really gave the first idea that there should be such a thing to study as permutations of things.
So what I'm saying is that the modern idea of this thing called a group, which is it's incredibly important in modern algebra, was really somehow invented in this idea. OK, we've mentioned the magic word is called a group. So every first year, mathematical and undergraduates, the world's over probably will learn about a group of what a group is. And this notion was first introduced as a series in England. What, Benjamin, can you explain to our listeners what a group is and how how it used to be?
Yeah, I think, you know, you can certainly try and gain. So Gaulois himself and other mathematicians at the time, he was just beginning to develop the idea of a group would have essentially considered a group as what modern mathematicians would have thought of as a subgroup of the permutation group. So let's try and explain what that is. A permutation, which we've actually mentioned a few times in terms of the roots, basically just means swapping things around.
So, for example, I might have three groups say I label them one, two and three. And a permutation of this would just be, for example, I could swap one and two and not swap three. So that that's a permutation or I could send one to two, two to three and three back to one again. And this is also a permutation. So the important thing is that everything is sent to something else, but you don't send two things to the same thing.
So one and two can't both be sent to three shuffling around like a deck of cards. Yeah, exactly. Yeah. Or for example, I mean, the classic kind of thing that maybe people are very familiar with is something new, for example, a Rubik's Cube. So there are lots of different ways that a Rubik's Cube can look. You want to obviously make it look so that all of its sides are the same colour. But of course, you can keep kind of turning it and make it look like many different configurations.
And this is basically something to do with how many permutations there are of the way a Rubik's Cube could look. And therefore, in some sense, quite, you know, in terms of one very naive way that you could try and think why group theory is useful. If you want to say solve a Rubik's Cube, you might want to therefore try and understand how many different permutations there are of the way it can look.
And by understanding how you get from it's kind of the state we want it to be to this weird state, I can try and get it back from this weird state to the state I want it to be. So by understanding permutations, I can try and solve, for example, a Rubik's cube and the kind of important I mean, there are a number of kind of abstract properties that a group is meant to satisfy.
But really for the purposes of I think for this podcast, it's fine for our readers really just to think of a group as some subset of permutations. So, for example, we talked about permutations of the numbers one, two and three. So it turns out there are six possible permutations of this, but you might not necessarily want to consider all six of them. So you might just want to consider a permutation which doesn't do anything.
So everything stays the same. And the permutation which switches one and two, and that's all the permutations you want to consider. So I think it's probably OK for our listeners just to think of a group as some kind of permutation of some object, of some set of things.
And the the insight, as Chris explained for us, is that the structure of the ways you can swap these roots of the polynomial is somehow intimately connected with whether you can solve it by radical's, which is this extraordinary, bizarre. So it's very important, in fact, to define exactly what we mean by swapping the roots in this example so that in what we just said, we said if you don't want two and three, then you will have to swap them and so on.
But the way this gets restricted and the reason you wouldn't use the whole permutation group is because you're only allowed to swap certain groups. And the rule is as follows. You might notice that, in fact, when you start with some polynomial, sometimes it can be broken down into smaller polynomials. For example, if I start with the polynomial X squared minus one, then.
You can see that, in fact, this is equal to the polynomial X plus one times X minus one, if you just multiply things out, you'll see that the the X terms kastle. So if you start with some polynomial, then one thing you can do is just break it down. And these things are equal to the factors of the irreducible factors. And the rule for when you can swap roots around is when they belong to the same irreducible factor.
So if I start with some polynomial same degree for and I split it up into some factors, maybe a degree to factor in another degree to factor, then this naturally gives the roots and it splits the reason to two collections. And within each collection I'm allowed to permit them however I like. So say we label them one, two, three, four. I could swap for one and two.
I could stop the three and four, but I can't swap those x. Yeah, no, that's I guess I mean in some sense the situation is a tiny bit more complicated because as Chris mentioned, it's important. The one we start with is polynomial. We break it down as kind of into small bits as we can so we can see which groups are allowed to be swapped with each other. Now, it does so happen that, in fact, the permutations which are allowed are in fact sensitive to the coefficients of the polynomial.
And that might seem quite surprising in some sense. But if there's any hope at all that trying to find these, you know, when is a polynomial solvable by radicals using group theory? Well, we know that these radical these polynomial equations are meant to be sensitive to the coefficients that the ultimate goal was to find roots in terms of coefficients.
So in some sense, it's good that the group theory is sensitive to what the coefficients were, because if it wasn't, that would somehow mean that, you know, my finding the general equation, which was meant to be sensitive to the coefficient of the group there, isn't sensitive to the coefficients. Then there's some kind of disconnect between the two methods.
Think I'm trying to get at is the fact that if you have an original polynomial degree five, the longer you can be as five and it can also be C five and everything in between. So I think that the kind of important thing that listeners should take away is that you break things down into small components as you can go irreducible factors and then some subset of the permutation of each of the roots of the irreducible factors is then allowed.
But exactly what subset you have to determine and insensitive to the coefficients of the polynomial themselves. Okay, so that was five minutes on group theory I think is incredible. Again, returning to go, the man he's submitted this paper to the PRISE that got lost admits it again to be refereed for a journal who didn't submit this paper to. So this is in 1831 and he submitted it to prosecute and so on.
In fact, invited him to submit this paper. But because perhaps he felt bad for losing the PRISE submission. And this was also rejected, I think. Yeah. So this is January 31 and by July, the referee report comes back. I was in prison at this time for him being arrested on Bastille Day for Revolutionary Goings on and rejects the paper summarising 31. So what happens in gavel's life that leads him to such an early death?
Well, so obviously he's incredibly disappointed and I think also angry by his the rejection of the second rejection. I guess his paper, the first rejection was a big loss of rejection. So in the meantime, he had been, I think, was in a nursing home due to a cholera outbreak, which was being experienced in France at the time. So this was he was transferred to this nursing home in early 1832, and he actually continued to pursue his research while at the nursing home.
He was working on revising the memoir rejected by Pakistan. And also, I mean, this is having had I thought a year or so of not producing so much, not know exactly due to being in prison in the end, which is perhaps not the best environment for any kind of original research. But he also so it's around this time that it appears that he'd found a love interest of some kind after he left the nursing home.
He was then challenged to a duel. And it's quite unclear whether this trial was due to his kind of revolutionary involvement, whether this was about his Republicanism or whether this was due to his kind of his love interest and whether this well, who why he was a former lover. I mean, there are two sources for what the issue might be about. And they say different things that we just don't know exactly what was the letter to his parents?
And one was a letter to his great friend Chevallier. But we're not sure. Yeah, well, what would you say to your parents? Love, interest or revolutionary folly wouldn't get in the jewel. But do you have any discussion about how does Chevallier come into our story? Yes, Chevallier is actually very important in publicising Gaulois. Workingman's essentially is one of Gallas friends and Gaulois in a letter to Chevallier.
This this letter we just mentioned before, the jewel says, please, you know, if anything happens, can you try and get my work to Gousse Galson Jacqui. Jacqui being one of the winners of the prise that she submitted to Chevallier, it takes us one year after to do this, but has a slightly unusual way doing this, which is just to publish work without actually mentioning this.
It seems so. Gasso Jacqui, we have been so this is now very famous letter, the testamentary letter written on the eve before that you know that you will go out got shot and he died in hospital two days later. So this is as ill fated, a romantic hero as you can choose to get. But yes, sorry is continually interrupted. So Chevallier takes have to publish Gower's work, but it doesn't seem to be noticed at all by Gousse or Giacobbe, who were the original intended recipients of glasswork.
Let us have our final article, chunk of the podcast now. Undergraduate students learning about Gaulois theory. Today we'll learn this thing called the Gaulois correspondence. There's not a thing that's so present in Gallas. What precisely? That is how it was interpreted later. Is that. Yes, yeah. That that's a fair assessment. So we have a lot of this have a lot of modern theory undergraduates, and I'm involved in teaching, as is Chris Holford teaching undergraduates here at Oxford.
The Gaulois theory you learn would have been quite alien to Gore himself. And somehow this is a fairly common mathematical theme where people get topics named after them that they may have were originally connected with, but they might have been quite surprised by the directions that they've taken. But yeah, so the most important result for the Gaulois theory that people are taught today at university is this thing called the Gaulois correspondence.
And we've mentioned group theory about how integral to Gaulois work is these kind of subgroups or subsets of permutations. So we might not be able to allow all the permutations of the roots, but we allowed some of them. And it's very important this what subset this is associated to each polynomial. And essentially what the goal of correspondence does is it relates different subsets of these permutations to something to something called different subfields.
And these will, I think, talk about exactly what a field is and what a subfield is in a set. But essentially this is relating to subgroups of permutations in some way to the roots of the polynomial. And therefore, you can see how this was quite important in terms of defining the formula because it gives an explicit relation between the subsets or subgroups and the roots of the polynomial itself. Well, so let's just go and mention what it feels. So it's another abstract algebraic object.
Absolutely. So a field is just any number system where you're allowed to add things, subtract things, but also divide by non-dairy everything's. Obviously, you can't abide by the thing I remember from school. So if I'm thinking of an example, I guess I've read the numbers on the number line or real numbers. That's exactly that would be a field. So if you take some number like two, something like PI, multiply them together, you get to pie.
This is also in the real numbers. And if you take to pie and you divide by you divide two by pi, get two over pi. This is also a real number. And so the real numbers are a field. But I guess the whole numbers, the integers are not a field because I can't define exactly. So if I take two and I take one, I try and divide one by two, I get half, which is not a whole number.
But the role of fixing this. Yeah. So so all the went wrong with taking the whole numbers is that you can't divide by whole numbers. But what if we just say, OK, well let's try and find the smallest thing which does include all these divisions. So what if we just say, OK, well, he doesn't need to have a half as well, you need to have a third, a quarter and so on, but you also need like five over two and so on.
And then you notice that what you're describing it is all fractions with whole number divided by Honolua. So all these factors are the rational numbers so that there's another field. But they were more exotic versions of these. Yes, exactly.
And sometimes what will or it's integral to our correspondence is the idea that as Chris says, so what essentially the rational numbers are or just fractions, is it saying if I want to have the whole numbers but I want to turn them into a field, I have to be able to divide by them. And the rational numbers is essentially the field that you get created from your positive integers.
And was integral to Gauloise and the goal correspondance is that you take your rational numbers, which we've decided as a failed, but I also want to be allowed to have certain routes of this polynomial, which may no longer be rational. And therefore, I need to essentially take the smallest field, which contains my rational numbers and the route. So, for example, if I have one which is saying that the square root of two, then I have to be able to add multiples of the square root of two to itself.
So I have to get any multiple the square root of two a.m. to divide by the square root of two and so on. So you can see how essentially the procedure for coming up with the rational numbers or just fractions can be generalised by kind of just adding another thing. We have to be allowed to have all divisions and all additions with. And this is in some sense the kind of the fields that will be interested in it.
Well, so let us as the climax of our discussion of this abstract machinery nailed down this correspondant. So we have on the one hand, these certain allowed shuffling of the roots of this equation is permutation. Yes. And then on the other hand, we have the field that comes out of this operation. The bend is described as I have my equation, say X squared minus two, and then I have the root of that equation, which is the square root of two.
And then I grow the field out of that. And both of the fractions, the goal correspondence is what type of connexion between this field objects I have on the one hand and this permutation object on the other, if we consider the permutations first. So let me give an example. So suppose I have roots, Alpha, Beta and Gamma and suppose I'm allowed to permit them however I like so I can take alpha switchfoot.
Peter and I can keep it fixed. I can switch out gamma beta fixed then to any one of these subsets of these permutations, which is consistent. I can associate a field and the field is going to be contained in the field, which contains all the roots, part of the philosophy of what is you want to have the minimum field that you can to contain everything.
So of course, one one could just move to maybe not, of course, but one could just move to the complex numbers, which would have all of your roots in it, but would also have lots of extra things that you don't need. So perhaps you wanted to have a square root of two, but the complex numbers also have a square root of three. You didn't need this, so you don't want it. So what you do is you you go to a small field which contains all the roots that you wanted.
And then this connexion is between a subgroup of your permutations and a subfield of this field there in direct one to one direction. And then as a final comment, how does finding these subfields corresponding to the subgroups help to solve the equation, which is ultimately what we are trying to do? Sure.
Well, so it turns out the same as Chris mentioned, these if I for example, in the degree free case, if we're allowed to have all permutations, then one of the permutations would just be swapping, say, the first and second route around and fixing the third. And therefore, the subfield this is corresponding to is when I take the rational numbers. And just to join this third route, because obviously we said this permutation fixed the third route.
And therefore, if I just take a field made by the rational numbers and adding my third route on, then this permutation will also should fix everything in this field. And this is how the correspondence is meant to work. Now, what's the Ghawar? So how Gower's results interpreted nowadays is essentially to do with what type of group? The whole permutations, all the permutations are. So there's various different types of groups.
And the crucial type that we're going to be interesting in is something called soluble, which sounds very suggestive. And we're trying to solve equations. And it's these types of groups that allow us to come up with equations for solving polynomial. So, for example, we said that we know that we can solve a quadratic.
We said that you can solve a Kubek in a quartette. And the reason for this is that, for example, in the Quartette case, there were the most permutations that we could have if all the permutations of the numbers one, two, three and four. And if you count them, there were twenty four of these. And it turns out that this group is a soluble group.
Now, it's probably not worth getting into what the technical definitions of this means, but similarly, all the permutations of one, two, three is also a soluble group and the permutations of just one and two is also a soluble group. However, when you go into the degree five case, in some cases you might want to have all permute.
Haitians of one, two, three, four and five, and this group is no longer soluble, and it's this connexion which again is not trivial to establish, that shows why a degree five polynomial, indeed a degree six and higher cannot always be solved by radicals. But of course, as we've kind of suggested, sometimes they could be because it may be that your degree five polynomial, it's associated Ghawar Group. This group of permutations might not be all the permutations of one, two, three, four and five.
It might be a smaller group and this might be soluble. So it's sensitive to the specific coefficients and thus kind of determines what this group of allowable permutations is. The change in abstracts algebraic structure when we go from four to five on the group theory side is the reason why five is exactly.
And I think what this maybe indicates is we talked about how, you know, surprising maybe this result would have been, particularly, say, in the hundreds where people were coming up with a Kubek and a quartette formula.
And it's just because well, given I've come up with a quadratic cubic and acoustic formula, surely I should be able to come up with a formula because it seems to be fitting a pattern and what the kind of point of abstract mathematics is, it's allowing us to kind of see the situation in a different light.
And when we connect this problem to group theory and you consider, you know, what groups are soluble is not to is fairly easy to show, the group of permutations of one, two, three, four and five is not soluble. And therefore suddenly this isn't quite so surprising anymore because there's a very obvious change in the group theory structure between degree four and degree five, which you don't see if you just look at polynomials themselves.
And this is in some kind of advert for why abstract mathematics is actually has a point to it. It allows you to better solve the original problem that you were trying to. I couldn't agree more. In the final minutes of the show, we've mentioned Chevallier who took out US papers and published them initially, ten years later gave them to Louisville, who in the 1940s began to begin the real politicisation of Galileo's work, which was then taken up for the rest of the nineteenth century.
What would you say are the most important applications of Gallas work and maybe more importantly, his ideas? Tsoukalas work has been incredibly influential across abstract algebra and just in terms of the sheer number of questions that this gives you more access to. So, for example, number three, my area of research and in fact all of our possessions, one big area of number three is all about can you solve certain equations?
So not necessarily just polynomial equations, but maybe equations in two variables. So instead of just execute plus one equals zero, you want to know what are the solutions to maybe Y squared equals execute plus one two variables and understanding these kind of things from different viewpoints, which is what Gaulois gives us access to. If we look at absolute algebra is just incredibly, incredibly useful. But outside number theory, I mean, has abstract algebra had applications in other areas?
Yeah, I mean, it's an absolute algebra has just become its own subject. I mean, you know, there are plenty, you know, the whole research on here, which does, you know, algebra or representation theory, which are both offshoots of maybe the stuff that Gaulois started researching all those hundreds of years ago. Almost all areas of pure mathematics now involve some form of abstract algebra and not just pure mathematics, applied mathematics as well.
And that's why we talked a lot about how when First-Year mathematicians come here. What group theory, though, that there's a reason why you learn group theory in first year, and it's because it underpins all of us so much the mathematics that you do later on. So pretty much any area of mathematics that we could mention will have kind of important one.
In particular, any theoretical physicist, you know, a huge amount of group theory because it underpins all the chemistry that chemistry, these permutations or symmetries are also important in studying the structures. And it all started at Gallo. Well, we're out of time, but thank you very much for bringing us through so much during the time. And thank you for listening.