1 00:00:00,720 --> 00:00:05,340 Hello, Everest, Gaulois, jimm and revolutionary, 2 00:00:05,340 --> 00:00:12,450 but it says on a commemorative French stamp made in honour of one of the most enigmatic and unusual mathematicians in history, 3 00:00:12,450 --> 00:00:18,480 he lived a very short life in the first half of the 19th century, dying when he was just 20 years old. 4 00:00:18,480 --> 00:00:26,640 But during that brief span, he produced work that, after his death, would go on to revolutionise mathematics in the years to come. 5 00:00:26,640 --> 00:00:31,110 Today, we will not only discuss the extraordinary details of his remarkable life, 6 00:00:31,110 --> 00:00:36,450 but also give a flavour of the kind of mathematics that his ideas gave rise to. 7 00:00:36,450 --> 00:00:43,380 With me to discuss Everest I working his life are Chris Nichols, a third year student in maths from Baylon College, 8 00:00:43,380 --> 00:00:48,420 and Benjamin Green, a final year student and lecturer at Baylor College. 9 00:00:48,420 --> 00:00:53,070 Thank you very much for joining me. I think if I look at the plan that we've made together correctly, 10 00:00:53,070 --> 00:01:00,240 we're going to alternate between some vague details of his mathematics, but also the details of his life. 11 00:01:00,240 --> 00:01:06,240 Chris, when was Gaulois born? Can you tell us a bit about his his upbringing and where he came to mathematics? 12 00:01:06,240 --> 00:01:17,430 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. 13 00:01:17,430 --> 00:01:22,800 So it was quite surprising when he developed this philosophy taught by his mother until about the age of 12 14 00:01:22,800 --> 00:01:31,140 and his father was eventually to become the mayor of this town when Gaulois does eventually go to school. 15 00:01:31,140 --> 00:01:35,940 Is he a successful people or would his teachers think of him as teachers? 16 00:01:35,940 --> 00:01:41,620 Aren't actually that impressed sometimes, although they're their comments that are not always that consistent. 17 00:01:41,620 --> 00:01:46,590 For example, one teacher actually describes him as singular, bizarre, original and closed. 18 00:01:46,590 --> 00:01:50,380 But in fact, it's exactly this originality that we remember him for. 19 00:01:50,380 --> 00:01:56,040 So, yes, I think the word it's a wonderful word, singular, because if you read a lot of, say, Arthur Conan Doyle, 20 00:01:56,040 --> 00:02:02,100 Sherlock Holmes books, he's singular to me like an odd occurrence, which is certainly never used in the English language now. 21 00:02:02,100 --> 00:02:07,860 So whether being singular is good for a people is certainly certainly up for debate, I guess. 22 00:02:07,860 --> 00:02:15,720 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. 23 00:02:15,720 --> 00:02:22,590 And whether he meant it or not, he discovers mathematics when he was about 14, 15. 24 00:02:22,590 --> 00:02:29,400 And this has a huge effect on him. And that's something that his teacher said about his mathematical ability at that time. 25 00:02:29,400 --> 00:02:38,310 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. 26 00:02:38,310 --> 00:02:42,480 And I think it would be best for him if his parents had allowed him to study nothing but this. 27 00:02:42,480 --> 00:02:47,190 He's wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments. 28 00:02:47,190 --> 00:02:53,700 So it sounds to there that he said if you found his message, he sent it, went for it. 29 00:02:53,700 --> 00:02:58,020 Yeah. And I think at this time, he's reading a lot of mathematics books that are that are out there. 30 00:02:58,020 --> 00:03:02,850 But a lot of the things that he's interested in actually just aren't very well developed at this time. 31 00:03:02,850 --> 00:03:05,940 So what he eventually becomes famous for is his work in algebra. 32 00:03:05,940 --> 00:03:11,130 But a lot of the textbooks that really aren't so well developed as a modern student would seem to me, 33 00:03:11,130 --> 00:03:16,110 if you read a modern textbook on algebra, it would be very, very different to what you would get. 34 00:03:16,110 --> 00:03:24,150 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. 35 00:03:24,150 --> 00:03:29,510 Well, I think a lot of mathematics then was developed to try and solve quite practical questions. 36 00:03:29,510 --> 00:03:35,310 So like people like Watson or Fourier, they have famous mathematical results associated with them, 37 00:03:35,310 --> 00:03:42,690 but which were often used or developed to try and solve practical mathematical problems and also practical real world problems, 38 00:03:42,690 --> 00:03:44,460 which had a mathematical angle. 39 00:03:44,460 --> 00:03:52,800 And what Ghawar himself became famous for, although not during his lifetime, many years after his death, was for much more abstract mathematics. 40 00:03:52,800 --> 00:04:00,690 And that is essentially taking a question which may be quite natural and trying to look at it far removed from the original problem. 41 00:04:00,690 --> 00:04:07,620 And as Chris mentioned, a kind of algebra as a modern mathematics student would learn about it would have been very, 42 00:04:07,620 --> 00:04:10,920 very different from the way the GAO or even people, you know, 43 00:04:10,920 --> 00:04:16,200 say, 10, 20 years after his death would understood algebra wasn't properly developed, 44 00:04:16,200 --> 00:04:21,960 maybe until the late eighteen hundreds in the manner that most students now would consider it. 45 00:04:21,960 --> 00:04:30,060 Exactly. If I recall correctly in was it eighty eight when Fourier introduced the phrase transform as we now know it, 46 00:04:30,060 --> 00:04:37,320 which is a major method of mathematical analysis, it was to to help solve a huge equation and understand heat propagation in body. 47 00:04:37,320 --> 00:04:44,910 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. 48 00:04:44,910 --> 00:04:50,820 Let us now do our first little stint of some of the actual mathematical content of Galileo's work, 49 00:04:50,820 --> 00:04:59,830 then mentioned to us that Gauloise work within algebra in particular is involved with solving equations and a particular kind of equations called. 50 00:04:59,830 --> 00:05:04,680 Polynomial equations, and that's a very long word, but Chris, that's not such a scary concept. 51 00:05:04,680 --> 00:05:08,370 It's very similar to what many of our listeners would have studied at school. Absolutely. 52 00:05:08,370 --> 00:05:17,820 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. 53 00:05:17,820 --> 00:05:22,200 And where X is some unknown. Exactly. So actually some unknown quantity. 54 00:05:22,200 --> 00:05:28,980 And I guess you can think of it as it's a formula that you can substitute some value into if you wanted to. 55 00:05:28,980 --> 00:05:36,750 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. 56 00:05:36,750 --> 00:05:43,950 And the thing that Gauloise really interested in is what kind of solutions there are to this thing equalling zero, 57 00:05:43,950 --> 00:05:47,910 these solutions called the roots of the polynomial. 58 00:05:47,910 --> 00:05:53,880 So I think at school, people may be familiar and maybe heated this quadratic equations. 59 00:05:53,880 --> 00:05:59,680 This is where you have A squared. So X squared plus three, X plus one equals zero. 60 00:05:59,680 --> 00:06:07,110 Exactly. But but you're saying but the interesting question is when you have maybe X cubed extra before. 61 00:06:07,110 --> 00:06:10,470 Exactly. So. So as as maybe you remember from school, I mean, not you, 62 00:06:10,470 --> 00:06:19,440 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, 63 00:06:19,440 --> 00:06:26,160 where A and B is some number. And the idea is that suppose you take some equation like X squared plus two 64 00:06:26,160 --> 00:06:30,900 equals one and you want to know what values of X can give a solution to this. 65 00:06:30,900 --> 00:06:37,350 So what values of X you can put in that'll will give X workers to express one equal to zero? 66 00:06:37,350 --> 00:06:44,370 And there's even a formula that tells you exactly when you can do this. So you can consider that case completely understood. 67 00:06:44,370 --> 00:06:47,850 But when you go to something like a cubic equation, which is something category three, 68 00:06:47,850 --> 00:06:53,070 maybe cube plus two X squared plus X plus one, then it gets a little bit harder. 69 00:06:53,070 --> 00:06:57,900 But people can still come up with some formula to solve this. 70 00:06:57,900 --> 00:07:08,460 Then when was this Kubic formula discovered? This was done by Italian mathematicians and the 1400's, so goodnews method is the standard method. 71 00:07:08,460 --> 00:07:12,660 Now, at least for solving a Kubic equation. I believe that was actually published by code, 72 00:07:12,660 --> 00:07:21,720 developed by him basically gives a false formula very similar to the quadratic equation formula that some of our listeners might remember, 73 00:07:21,720 --> 00:07:31,530 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, 74 00:07:31,530 --> 00:07:39,000 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. 75 00:07:39,000 --> 00:07:39,460 Exactly. 76 00:07:39,460 --> 00:07:48,360 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. 77 00:07:48,360 --> 00:07:53,100 So of course there are those four of those and the quadratic case, there are three of those. 78 00:07:53,100 --> 00:08:01,920 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. 79 00:08:01,920 --> 00:08:06,120 So again, you got a slightly more complicated expression now, 80 00:08:06,120 --> 00:08:15,030 depending on the five coefficients which appear in a general degree form four a.m. and that and the thing about these equations is, 81 00:08:15,030 --> 00:08:20,970 you know, once mathematicians had found a degree for a degree, for one, it was a natural question to ask. 82 00:08:20,970 --> 00:08:26,430 Well, if you give me a degree five polynomial, is there a similar formula for this and so on. 83 00:08:26,430 --> 00:08:32,310 So a formula just involving the coefficients, I guess there are six coefficients now is a X to the five. 84 00:08:32,310 --> 00:08:37,020 Must be X before the formula, just involving those coefficients. 85 00:08:37,020 --> 00:08:41,580 That will give me a solution to this equation in zero. 86 00:08:41,580 --> 00:08:46,560 The exact statement about what we mean by a formula is slightly complicated. 87 00:08:46,560 --> 00:08:55,410 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. 88 00:08:55,410 --> 00:08:59,160 Now, similarly, in the cubic formula, there are additions, division, 89 00:08:59,160 --> 00:09:06,810 multiplication going on and there are square and cube roots and similarly the degree four equation for groups. 90 00:09:06,810 --> 00:09:11,490 So basically what they were kind of hoping for was in the degree five case, 91 00:09:11,490 --> 00:09:17,340 and if you kept going up that you'd be able to just get a formula involving basic addition, 92 00:09:17,340 --> 00:09:22,500 subtraction, multiplication and division and also taking root. 93 00:09:22,500 --> 00:09:25,770 So square root, key root for fruits and so on. 94 00:09:25,770 --> 00:09:30,620 And that's now. Exactly. And that and that would be how you could state all the solutions. 95 00:09:30,620 --> 00:09:39,060 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, 96 00:09:39,060 --> 00:09:47,340 that for any 50 degree polynomial, there is some formula that takes the coefficients and gives you a root. 97 00:09:47,340 --> 00:09:52,050 Chris, so Gallo worked in this question and this is, you know, by the early 19th century. 98 00:09:52,050 --> 00:09:54,480 So it's been unsolved for a very long time. 99 00:09:54,480 --> 00:10:02,790 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. 100 00:10:02,790 --> 00:10:03,030 Now, 101 00:10:03,030 --> 00:10:12,570 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. 102 00:10:12,570 --> 00:10:17,310 So it may be that for a particular degree, five four, you can find the roots. 103 00:10:17,310 --> 00:10:24,660 For example, if I take the polynomial X to the five minus one equals zero, then certainly one root is X equals one. 104 00:10:24,660 --> 00:10:27,870 One of the five. That's about a one 1.0. 105 00:10:27,870 --> 00:10:35,820 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, 106 00:10:35,820 --> 00:10:42,570 then there's no formula just involving roots of some degree and coefficients in this polynomial. 107 00:10:42,570 --> 00:10:48,690 Which is amazing because, you know, you have this natural idea where you say, OK, well, I can solve every two problems. 108 00:10:48,690 --> 00:10:53,760 We can eventually solve degree three polynomials and then degree four. And it really looks like this pattern should continue. 109 00:10:53,760 --> 00:10:58,320 I mean, there should be some way of rearranging operations, taking roots to get the solution. 110 00:10:58,320 --> 00:11:03,300 But this amazing result that is not possible is extraordinary and very, very surprising. 111 00:11:03,300 --> 00:11:10,080 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. 112 00:11:10,080 --> 00:11:15,190 Let us come back to Galois and his life. So we've left him at the end of school. 113 00:11:15,190 --> 00:11:21,660 Then he tries to get into the Ecole Polytechnique, which is the premier school in Paris at the time. 114 00:11:21,660 --> 00:11:25,260 How does that go? Well, it doesn't go very well for him. 115 00:11:25,260 --> 00:11:30,600 He was rejected from the whole technique, which was something that he was obviously very disappointed about. 116 00:11:30,600 --> 00:11:35,220 He, as we heard kind of earlier, he developed a real passion for mathematics. 117 00:11:35,220 --> 00:11:39,380 And the technique was founded by a mathematician in the late seventeen hundreds. 118 00:11:39,380 --> 00:11:45,480 So it was certainly the pre-eminent place in Paris and indeed one of the best places in the world to be studying mathematics, 119 00:11:45,480 --> 00:11:50,880 because in France there was quite a tradition of good mathematics around. 120 00:11:50,880 --> 00:11:57,960 And therefore, yeah. So certainly for for Gaulois it was incredibly disappointing to have been rejected from the school. 121 00:11:57,960 --> 00:12:05,530 So he instead had to go to. Ecole Normal, or I think it was called something slightly different at the time, 122 00:12:05,530 --> 00:12:10,180 but that's what it's known as now, which I think you mentioned was a teacher training. 123 00:12:10,180 --> 00:12:14,860 Yeah, it was the place where if you were going to go on to become some kind of secondary school teacher, 124 00:12:14,860 --> 00:12:24,340 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. 125 00:12:24,340 --> 00:12:28,440 Yeah. So it was very much, you know, still confined within that same place. 126 00:12:28,440 --> 00:12:31,810 So he was very happy to be there. But he tried a couple of times. 127 00:12:31,810 --> 00:12:39,430 They're called Polytechnique. And yeah, there was a rather sad that happened in his life just before the second time. 128 00:12:39,430 --> 00:12:45,730 Yeah. His father committed suicide in 1829. I think a few weeks after the death of his father. 129 00:12:45,730 --> 00:12:51,550 He actually tried again and was rejected again. And I think also the the is the suicide of his father, 130 00:12:51,550 --> 00:12:56,080 I think was related to we mentioned that his father was a matter of his local town and it 131 00:12:56,080 --> 00:13:01,600 was related to the various political upheavals that were going on in France at the time. 132 00:13:01,600 --> 00:13:10,630 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 133 00:13:10,630 --> 00:13:16,300 in the politics to remove the monarchy or anti monarchy that was going around in France at the time. 134 00:13:16,300 --> 00:13:22,330 And therefore, this might have pushed him even closer to being very tied into a political ideology. 135 00:13:22,330 --> 00:13:26,500 I mean, revolutions to a penny in France in the 19th century. 136 00:13:26,500 --> 00:13:37,600 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. 137 00:13:37,600 --> 00:13:42,310 But it is a time of great upheaval in parts of around this time. 138 00:13:42,310 --> 00:13:53,600 I mean, beloved light over lots of details. In early 1831, Gaulois submits a paper to the academy for submission to a prise. 139 00:13:53,600 --> 00:13:57,900 Chris, can you tell us a bit about what was in this paper and its history, its story? 140 00:13:57,900 --> 00:14:06,220 Absolutely. So so this paper is actually about the solubility of various equations in the way that we've discussed before, 141 00:14:06,220 --> 00:14:10,510 using brute sense and functions of the coefficients. 142 00:14:10,510 --> 00:14:16,040 But not many people actually liked this paper because the main statement is that. 143 00:14:16,040 --> 00:14:20,570 If you have a polynomial of prime degree, so, for example, 144 00:14:20,570 --> 00:14:25,520 it starts with the five and then it's about two things, or maybe it starts on the seven side of things, 145 00:14:25,520 --> 00:14:32,180 then it's solvable in this way, which we can refer to from now on by radicals, 146 00:14:32,180 --> 00:14:37,010 sold by radicals means solvable by a formula of the kind that Ben discussed earlier. 147 00:14:37,010 --> 00:14:49,400 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. 148 00:14:49,400 --> 00:14:56,090 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. 149 00:14:56,090 --> 00:15:05,240 Then the condition is that I need to be able to express Gamma as something like alpha squared plus beta, all of alpha. 150 00:15:05,240 --> 00:15:12,260 So it's some function like this intented, often beta. Technically, I guess I would call it a rational function in beta. 151 00:15:12,260 --> 00:15:20,240 But this is peculiar because we talked about trying to understand in terms of the coefficients of the polynomial, what the roots are. 152 00:15:20,240 --> 00:15:26,930 But Gallas rather perversely, he's saying, well, you can understand the roots if and only if the roots have this property. 153 00:15:26,930 --> 00:15:34,010 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. 154 00:15:34,010 --> 00:15:41,190 And to understand this thing, you need to first find the roots. Ibrahim Apriori, he submitted this paper for a prise at the academy. 155 00:15:41,190 --> 00:15:45,170 And what happened? What actually happened? I think the paper was lost. 156 00:15:45,170 --> 00:15:52,700 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, 157 00:15:52,700 --> 00:15:57,470 and he was involved in deciding he received it actually died at the time. 158 00:15:57,470 --> 00:16:04,040 And it's not quite known whether the fact that Galois manuscript was lost was connected to the fact that obviously there 159 00:16:04,040 --> 00:16:11,180 was this discontinuity in the procedure because the person in charge of the PRISE died during the decision making process. 160 00:16:11,180 --> 00:16:18,680 But this manuscript that Gaulois submitted was lost and in fact, the PRISE was given instead to Arbel and Jakob, 161 00:16:18,680 --> 00:16:25,650 which is in some sense quite a funny coincidence, given the Arbel, in fact, also did work very similar to ours. 162 00:16:25,650 --> 00:16:29,090 So this was, in fact, not for everyone to go on. 163 00:16:29,090 --> 00:16:38,360 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. 164 00:16:38,360 --> 00:16:45,590 It's also perhaps worth saying that Arbel himself had in fact just died in 18 and 29 at the age of 26. 165 00:16:45,590 --> 00:16:50,630 So there was quite a collection of young mathematicians dying prematurely in Europe at the time. 166 00:16:50,630 --> 00:16:55,820 There is something about trying to solve equations by radicals leads to early death, it seems. 167 00:16:55,820 --> 00:17:04,520 Yes, this is the inference. Okay, before we start to return to some of the methods that were used by setting this up, 168 00:17:04,520 --> 00:17:11,510 how it is Ghale was different from Able's and why are we making sure about Gaulois rather than able? 169 00:17:11,510 --> 00:17:16,280 How did Gaulois make a more fundamental contribution to Galileo's contribution? 170 00:17:16,280 --> 00:17:26,550 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, 171 00:17:26,550 --> 00:17:33,770 the first to actually make the link explicitly between whether a polynomial equation is solvable, 172 00:17:33,770 --> 00:17:39,210 by which I mean whether we can write down all the all the roots in terms of these radicals and. 173 00:17:39,210 --> 00:17:46,230 And permutations of the routes by which I mean take the routes and consider different ways in which you can rearrange them. 174 00:17:46,230 --> 00:17:51,630 So if I had the routes, Alpha, Beta and Gamma, then I could equally well consider the routes gamma, beta and alpha. 175 00:17:51,630 --> 00:17:56,590 So it's just different ways of arranging the routes. And Gallimard, the link between. 176 00:17:56,590 --> 00:18:05,590 The ways in which these routes can be rearranged and whether or not the equation is actually solvable in terms of radicals, 177 00:18:05,590 --> 00:18:12,310 there is like a Ruffini also read this kind of massive book on this subject, which is centred around, you know, read. 178 00:18:12,310 --> 00:18:21,910 But it's this explicit study of these shuffling of the routes and the routes first. 179 00:18:21,910 --> 00:18:29,260 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. 180 00:18:29,260 --> 00:18:36,730 And the reason this is interesting is because this paves the way for much of modern algebra in terms of studying things 181 00:18:36,730 --> 00:18:44,650 abstractly and really gave the first idea that there should be such a thing to study as permutations of things. 182 00:18:44,650 --> 00:18:49,750 So what I'm saying is that the modern idea of this thing called a group, 183 00:18:49,750 --> 00:18:56,290 which is it's incredibly important in modern algebra, was really somehow invented in this idea. 184 00:18:56,290 --> 00:18:59,770 OK, we've mentioned the magic word is called a group. 185 00:18:59,770 --> 00:19:09,070 So every first year, mathematical and undergraduates, the world's over probably will learn about a group of what a group is. 186 00:19:09,070 --> 00:19:14,170 And this notion was first introduced as a series in England. 187 00:19:14,170 --> 00:19:20,620 What, Benjamin, can you explain to our listeners what a group is and how how it used to be? 188 00:19:20,620 --> 00:19:27,160 Yeah, I think, you know, you can certainly try and gain. So Gaulois himself and other mathematicians at the time, 189 00:19:27,160 --> 00:19:32,110 he was just beginning to develop the idea of a group would have essentially considered a group 190 00:19:32,110 --> 00:19:37,930 as what modern mathematicians would have thought of as a subgroup of the permutation group. 191 00:19:37,930 --> 00:19:40,390 So let's try and explain what that is. 192 00:19:40,390 --> 00:19:47,290 A permutation, which we've actually mentioned a few times in terms of the roots, basically just means swapping things around. 193 00:19:47,290 --> 00:19:52,300 So, for example, I might have three groups say I label them one, two and three. 194 00:19:52,300 --> 00:19:58,570 And a permutation of this would just be, for example, I could swap one and two and not swap three. 195 00:19:58,570 --> 00:20:04,330 So that that's a permutation or I could send one to two, two to three and three back to one again. 196 00:20:04,330 --> 00:20:12,790 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. 197 00:20:12,790 --> 00:20:17,770 So one and two can't both be sent to three shuffling around like a deck of cards. 198 00:20:17,770 --> 00:20:18,730 Yeah, exactly. Yeah. 199 00:20:18,730 --> 00:20:24,880 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. 200 00:20:24,880 --> 00:20:28,750 So there are lots of different ways that a Rubik's Cube can look. 201 00:20:28,750 --> 00:20:32,680 You want to obviously make it look so that all of its sides are the same colour. 202 00:20:32,680 --> 00:20:37,870 But of course, you can keep kind of turning it and make it look like many different configurations. 203 00:20:37,870 --> 00:20:44,020 And this is basically something to do with how many permutations there are of the way a Rubik's Cube could look. 204 00:20:44,020 --> 00:20:51,460 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. 205 00:20:51,460 --> 00:20:53,860 If you want to say solve a Rubik's Cube, 206 00:20:53,860 --> 00:21:00,580 you might want to therefore try and understand how many different permutations there are of the way it can look. 207 00:21:00,580 --> 00:21:07,030 And by understanding how you get from it's kind of the state we want it to be to this weird state, 208 00:21:07,030 --> 00:21:11,890 I can try and get it back from this weird state to the state I want it to be. 209 00:21:11,890 --> 00:21:19,570 So by understanding permutations, I can try and solve, for example, a Rubik's cube and the kind of important I mean, 210 00:21:19,570 --> 00:21:23,290 there are a number of kind of abstract properties that a group is meant to satisfy. 211 00:21:23,290 --> 00:21:33,540 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. 212 00:21:33,540 --> 00:21:39,280 So, for example, we talked about permutations of the numbers one, two and three. 213 00:21:39,280 --> 00:21:46,000 So it turns out there are six possible permutations of this, but you might not necessarily want to consider all six of them. 214 00:21:46,000 --> 00:21:49,720 So you might just want to consider a permutation which doesn't do anything. 215 00:21:49,720 --> 00:21:56,080 So everything stays the same. And the permutation which switches one and two, and that's all the permutations you want to consider. 216 00:21:56,080 --> 00:22:04,750 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. 217 00:22:04,750 --> 00:22:08,350 And the the insight, as Chris explained for us, 218 00:22:08,350 --> 00:22:14,800 is that the structure of the ways you can swap these roots of the polynomial 219 00:22:14,800 --> 00:22:18,820 is somehow intimately connected with whether you can solve it by radical's, 220 00:22:18,820 --> 00:22:21,550 which is this extraordinary, bizarre. 221 00:22:21,550 --> 00:22:30,750 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, 222 00:22:30,750 --> 00:22:34,960 we said if you don't want two and three, then you will have to swap them and so on. 223 00:22:34,960 --> 00:22:42,700 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. 224 00:22:42,700 --> 00:22:45,310 And the rule is as follows. 225 00:22:45,310 --> 00:22:51,490 You might notice that, in fact, when you start with some polynomial, sometimes it can be broken down into smaller polynomials. 226 00:22:51,490 --> 00:22:56,500 For example, if I start with the polynomial X squared minus one, then. 227 00:22:56,500 --> 00:23:01,060 You can see that, in fact, this is equal to the polynomial X plus one times X minus one, 228 00:23:01,060 --> 00:23:05,110 if you just multiply things out, you'll see that the the X terms kastle. 229 00:23:05,110 --> 00:23:11,650 So if you start with some polynomial, then one thing you can do is just break it down. 230 00:23:11,650 --> 00:23:18,040 And these things are equal to the factors of the irreducible factors. 231 00:23:18,040 --> 00:23:25,090 And the rule for when you can swap roots around is when they belong to the same irreducible factor. 232 00:23:25,090 --> 00:23:29,830 So if I start with some polynomial same degree for and I split it up into some factors, 233 00:23:29,830 --> 00:23:39,340 maybe a degree to factor in another degree to factor, then this naturally gives the roots and it splits the reason to two collections. 234 00:23:39,340 --> 00:23:43,300 And within each collection I'm allowed to permit them however I like. 235 00:23:43,300 --> 00:23:47,440 So say we label them one, two, three, four. I could swap for one and two. 236 00:23:47,440 --> 00:23:51,880 I could stop the three and four, but I can't swap those x. 237 00:23:51,880 --> 00:23:59,620 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. 238 00:23:59,620 --> 00:24:01,060 The one we start with is polynomial. 239 00:24:01,060 --> 00:24:08,890 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. 240 00:24:08,890 --> 00:24:16,600 Now, it does so happen that, in fact, the permutations which are allowed are in fact sensitive to the coefficients of the polynomial. 241 00:24:16,600 --> 00:24:19,270 And that might seem quite surprising in some sense. 242 00:24:19,270 --> 00:24:26,560 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? 243 00:24:26,560 --> 00:24:31,210 Well, we know that these radical these polynomial equations are meant to be sensitive to 244 00:24:31,210 --> 00:24:36,130 the coefficients that the ultimate goal was to find roots in terms of coefficients. 245 00:24:36,130 --> 00:24:44,470 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, 246 00:24:44,470 --> 00:24:52,330 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. 247 00:24:52,330 --> 00:24:55,460 Then there's some kind of disconnect between the two methods. 248 00:24:55,460 --> 00:24:59,380 Think I'm trying to get at is the fact that if you have an original polynomial degree five, 249 00:24:59,380 --> 00:25:03,490 the longer you can be as five and it can also be C five and everything in between. 250 00:25:03,490 --> 00:25:10,210 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 251 00:25:10,210 --> 00:25:19,090 can go irreducible factors and then some subset of the permutation of each of the roots of the irreducible factors is then allowed. 252 00:25:19,090 --> 00:25:25,690 But exactly what subset you have to determine and insensitive to the coefficients of the polynomial themselves. 253 00:25:25,690 --> 00:25:30,040 Okay, so that was five minutes on group theory I think is incredible. 254 00:25:30,040 --> 00:25:31,530 Again, returning to go, 255 00:25:31,530 --> 00:25:43,750 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. 256 00:25:43,750 --> 00:25:47,950 So this is in 1831 and he submitted it to prosecute and so on. 257 00:25:47,950 --> 00:25:55,660 In fact, invited him to submit this paper. But because perhaps he felt bad for losing the PRISE submission. 258 00:25:55,660 --> 00:25:59,920 And this was also rejected, I think. 259 00:25:59,920 --> 00:26:05,650 Yeah. So this is January 31 and by July, the referee report comes back. 260 00:26:05,650 --> 00:26:14,620 I was in prison at this time for him being arrested on Bastille Day for Revolutionary Goings on and rejects the paper summarising 31. 261 00:26:14,620 --> 00:26:20,680 So what happens in gavel's life that leads him to such an early death? 262 00:26:20,680 --> 00:26:27,550 Well, so obviously he's incredibly disappointed and I think also angry by his the rejection of the second rejection. 263 00:26:27,550 --> 00:26:33,610 I guess his paper, the first rejection was a big loss of rejection. 264 00:26:33,610 --> 00:26:42,100 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. 265 00:26:42,100 --> 00:26:50,890 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. 266 00:26:50,890 --> 00:26:54,760 He was working on revising the memoir rejected by Pakistan. 267 00:26:54,760 --> 00:26:59,810 And also, I mean, this is having had I thought a year or so of not producing so much, 268 00:26:59,810 --> 00:27:09,340 not know exactly due to being in prison in the end, which is perhaps not the best environment for any kind of original research. 269 00:27:09,340 --> 00:27:18,940 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. 270 00:27:18,940 --> 00:27:27,160 He was then challenged to a duel. And it's quite unclear whether this trial was due to his kind of revolutionary involvement, 271 00:27:27,160 --> 00:27:35,350 whether this was about his Republicanism or whether this was due to his kind of his love interest and whether this well, 272 00:27:35,350 --> 00:27:41,200 who why he was a former lover. I mean, there are two sources for what the issue might be about. 273 00:27:41,200 --> 00:27:44,980 And they say different things that we just don't know exactly what was the letter to his parents? 274 00:27:44,980 --> 00:27:48,650 And one was a letter to his great friend Chevallier. But we're not sure. 275 00:27:48,650 --> 00:27:56,310 Yeah, well, what would you say to your parents? Love, interest or revolutionary folly wouldn't get in the jewel. 276 00:27:56,310 --> 00:28:00,330 But do you have any discussion about how does Chevallier come into our story? 277 00:28:00,330 --> 00:28:03,930 Yes, Chevallier is actually very important in publicising Gaulois. 278 00:28:03,930 --> 00:28:09,990 Workingman's essentially is one of Gallas friends and Gaulois in a letter to Chevallier. 279 00:28:09,990 --> 00:28:19,890 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. 280 00:28:19,890 --> 00:28:26,130 Jacqui being one of the winners of the prise that she submitted to Chevallier, it takes us one year after to do this, 281 00:28:26,130 --> 00:28:33,450 but has a slightly unusual way doing this, which is just to publish work without actually mentioning this. 282 00:28:33,450 --> 00:28:37,540 It seems so. Gasso Jacqui, we have been so this is now very famous letter, 283 00:28:37,540 --> 00:28:44,260 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. 284 00:28:44,260 --> 00:28:50,550 So this is as ill fated, a romantic hero as you can choose to get. 285 00:28:50,550 --> 00:28:54,300 But yes, sorry is continually interrupted. 286 00:28:54,300 --> 00:29:02,730 So Chevallier takes have to publish Gower's work, but it doesn't seem to be noticed at all by Gousse or Giacobbe, 287 00:29:02,730 --> 00:29:07,860 who were the original intended recipients of glasswork. 288 00:29:07,860 --> 00:29:13,020 Let us have our final article, chunk of the podcast now. 289 00:29:13,020 --> 00:29:19,140 Undergraduate students learning about Gaulois theory. Today we'll learn this thing called the Gaulois correspondence. 290 00:29:19,140 --> 00:29:24,470 There's not a thing that's so present in Gallas. What precisely? That is how it was interpreted later. 291 00:29:24,470 --> 00:29:32,550 Is that. Yes, yeah. That that's a fair assessment. So we have a lot of this have a lot of modern theory undergraduates, 292 00:29:32,550 --> 00:29:37,950 and I'm involved in teaching, as is Chris Holford teaching undergraduates here at Oxford. 293 00:29:37,950 --> 00:29:41,640 The Gaulois theory you learn would have been quite alien to Gore himself. 294 00:29:41,640 --> 00:29:48,180 And somehow this is a fairly common mathematical theme where people get topics named after them that they may have were originally connected with, 295 00:29:48,180 --> 00:29:52,440 but they might have been quite surprised by the directions that they've taken. 296 00:29:52,440 --> 00:29:52,860 But yeah, 297 00:29:52,860 --> 00:30:03,330 so the most important result for the Gaulois theory that people are taught today at university is this thing called the Gaulois correspondence. 298 00:30:03,330 --> 00:30:13,140 And we've mentioned group theory about how integral to Gaulois work is these kind of subgroups or subsets of permutations. 299 00:30:13,140 --> 00:30:17,850 So we might not be able to allow all the permutations of the roots, but we allowed some of them. 300 00:30:17,850 --> 00:30:23,970 And it's very important this what subset this is associated to each polynomial. 301 00:30:23,970 --> 00:30:29,430 And essentially what the goal of correspondence does is it relates different subsets 302 00:30:29,430 --> 00:30:34,080 of these permutations to something to something called different subfields. 303 00:30:34,080 --> 00:30:38,880 And these will, I think, talk about exactly what a field is and what a subfield is in a set. 304 00:30:38,880 --> 00:30:46,740 But essentially this is relating to subgroups of permutations in some way to the roots of the polynomial. 305 00:30:46,740 --> 00:30:51,960 And therefore, you can see how this was quite important in terms of defining the formula because it gives an 306 00:30:51,960 --> 00:31:00,000 explicit relation between the subsets or subgroups and the roots of the polynomial itself. 307 00:31:00,000 --> 00:31:04,620 Well, so let's just go and mention what it feels. So it's another abstract algebraic object. 308 00:31:04,620 --> 00:31:16,050 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. 309 00:31:16,050 --> 00:31:20,220 Obviously, you can't abide by the thing I remember from school. 310 00:31:20,220 --> 00:31:27,550 So if I'm thinking of an example, I guess I've read the numbers on the number line or real numbers. 311 00:31:27,550 --> 00:31:34,740 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. 312 00:31:34,740 --> 00:31:41,460 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. 313 00:31:41,460 --> 00:31:46,740 This is also a real number. And so the real numbers are a field. 314 00:31:46,740 --> 00:31:51,630 But I guess the whole numbers, the integers are not a field because I can't define exactly. 315 00:31:51,630 --> 00:31:56,910 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. 316 00:31:56,910 --> 00:32:04,380 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. 317 00:32:04,380 --> 00:32:11,250 But what if we just say, OK, well let's try and find the smallest thing which does include all these divisions. 318 00:32:11,250 --> 00:32:15,180 So what if we just say, OK, well, he doesn't need to have a half as well, 319 00:32:15,180 --> 00:32:19,530 you need to have a third, a quarter and so on, but you also need like five over two and so on. 320 00:32:19,530 --> 00:32:24,930 And then you notice that what you're describing it is all fractions with whole number divided by Honolua. 321 00:32:24,930 --> 00:32:29,730 So all these factors are the rational numbers so that there's another field. 322 00:32:29,730 --> 00:32:32,640 But they were more exotic versions of these. Yes, exactly. 323 00:32:32,640 --> 00:32:39,420 And sometimes what will or it's integral to our correspondence is the idea that as Chris says, 324 00:32:39,420 --> 00:32:43,020 so what essentially the rational numbers are or just fractions, 325 00:32:43,020 --> 00:32:49,440 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. 326 00:32:49,440 --> 00:32:55,510 And the rational numbers is essentially the field that you get created from your positive integers. 327 00:32:55,510 --> 00:33:02,770 And was integral to Gauloise and the goal correspondance is that you take your rational numbers, which we've decided as a failed, 328 00:33:02,770 --> 00:33:08,890 but I also want to be allowed to have certain routes of this polynomial, which may no longer be rational. 329 00:33:08,890 --> 00:33:14,500 And therefore, I need to essentially take the smallest field, which contains my rational numbers and the route. 330 00:33:14,500 --> 00:33:18,550 So, for example, if I have one which is saying that the square root of two, 331 00:33:18,550 --> 00:33:23,170 then I have to be able to add multiples of the square root of two to itself. 332 00:33:23,170 --> 00:33:28,870 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. 333 00:33:28,870 --> 00:33:34,240 So you can see how essentially the procedure for coming up with the rational 334 00:33:34,240 --> 00:33:39,220 numbers or just fractions can be generalised by kind of just adding another thing. 335 00:33:39,220 --> 00:33:43,210 We have to be allowed to have all divisions and all additions with. 336 00:33:43,210 --> 00:33:49,060 And this is in some sense the kind of the fields that will be interested in it. 337 00:33:49,060 --> 00:33:54,700 Well, so let us as the climax of our discussion of this abstract machinery nailed down this correspondant. 338 00:33:54,700 --> 00:34:03,110 So we have on the one hand, these certain allowed shuffling of the roots of this equation is permutation. 339 00:34:03,110 --> 00:34:11,680 Yes. And then on the other hand, we have the field that comes out of this operation. 340 00:34:11,680 --> 00:34:21,310 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. 341 00:34:21,310 --> 00:34:26,380 And then I grow the field out of that. And both of the fractions, 342 00:34:26,380 --> 00:34:35,590 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, 343 00:34:35,590 --> 00:34:40,120 if we consider the permutations first. So let me give an example. 344 00:34:40,120 --> 00:34:48,050 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. 345 00:34:48,050 --> 00:34:57,790 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. 346 00:34:57,790 --> 00:35:07,540 I can associate a field and the field is going to be contained in the field, which contains all the roots, 347 00:35:07,540 --> 00:35:14,920 part of the philosophy of what is you want to have the minimum field that you can to contain everything. 348 00:35:14,920 --> 00:35:22,090 So of course, one one could just move to maybe not, of course, but one could just move to the complex numbers, 349 00:35:22,090 --> 00:35:27,160 which would have all of your roots in it, but would also have lots of extra things that you don't need. 350 00:35:27,160 --> 00:35:34,540 So perhaps you wanted to have a square root of two, but the complex numbers also have a square root of three. 351 00:35:34,540 --> 00:35:42,610 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. 352 00:35:42,610 --> 00:35:53,470 And then this connexion is between a subgroup of your permutations and a subfield of this field there in direct one to one direction. 353 00:35:53,470 --> 00:36:02,440 And then as a final comment, how does finding these subfields corresponding to the subgroups help to solve the equation, 354 00:36:02,440 --> 00:36:05,020 which is ultimately what we are trying to do? Sure. 355 00:36:05,020 --> 00:36:13,360 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, 356 00:36:13,360 --> 00:36:18,910 then one of the permutations would just be swapping, say, the first and second route around and fixing the third. 357 00:36:18,910 --> 00:36:23,770 And therefore, the subfield this is corresponding to is when I take the rational numbers. 358 00:36:23,770 --> 00:36:29,380 And just to join this third route, because obviously we said this permutation fixed the third route. 359 00:36:29,380 --> 00:36:35,320 And therefore, if I just take a field made by the rational numbers and adding my third route on, 360 00:36:35,320 --> 00:36:38,800 then this permutation will also should fix everything in this field. 361 00:36:38,800 --> 00:36:43,450 And this is how the correspondence is meant to work. Now, what's the Ghawar? 362 00:36:43,450 --> 00:36:51,190 So how Gower's results interpreted nowadays is essentially to do with what type of group? 363 00:36:51,190 --> 00:36:55,360 The whole permutations, all the permutations are. 364 00:36:55,360 --> 00:36:57,580 So there's various different types of groups. 365 00:36:57,580 --> 00:37:04,660 And the crucial type that we're going to be interesting in is something called soluble, which sounds very suggestive. 366 00:37:04,660 --> 00:37:15,050 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. 367 00:37:15,050 --> 00:37:19,390 So, for example, we said that we know that we can solve a quadratic. 368 00:37:19,390 --> 00:37:26,440 We said that you can solve a Kubek in a quartette. And the reason for this is that, for example, in the Quartette case, 369 00:37:26,440 --> 00:37:32,560 there were the most permutations that we could have if all the permutations of the numbers one, two, three and four. 370 00:37:32,560 --> 00:37:38,350 And if you count them, there were twenty four of these. And it turns out that this group is a soluble group. 371 00:37:38,350 --> 00:37:42,820 Now, it's probably not worth getting into what the technical definitions of this means, but similarly, 372 00:37:42,820 --> 00:37:49,510 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. 373 00:37:49,510 --> 00:37:55,270 However, when you go into the degree five case, in some cases you might want to have all permute. 374 00:37:55,270 --> 00:38:05,590 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, 375 00:38:05,590 --> 00:38:13,750 that shows why a degree five polynomial, indeed a degree six and higher cannot always be solved by radicals. 376 00:38:13,750 --> 00:38:23,020 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. 377 00:38:23,020 --> 00:38:28,720 This group of permutations might not be all the permutations of one, two, three, four and five. 378 00:38:28,720 --> 00:38:32,200 It might be a smaller group and this might be soluble. 379 00:38:32,200 --> 00:38:40,660 So it's sensitive to the specific coefficients and thus kind of determines what this group of allowable permutations is. 380 00:38:40,660 --> 00:38:49,720 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. 381 00:38:49,720 --> 00:38:56,500 And I think what this maybe indicates is we talked about how, you know, surprising maybe this result would have been, 382 00:38:56,500 --> 00:39:01,090 particularly, say, in the hundreds where people were coming up with a Kubek and a quartette formula. 383 00:39:01,090 --> 00:39:06,880 And it's just because well, given I've come up with a quadratic cubic and acoustic formula, 384 00:39:06,880 --> 00:39:14,720 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, 385 00:39:14,720 --> 00:39:19,180 it's allowing us to kind of see the situation in a different light. 386 00:39:19,180 --> 00:39:26,440 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, 387 00:39:26,440 --> 00:39:31,000 the group of permutations of one, two, three, four and five is not soluble. 388 00:39:31,000 --> 00:39:34,780 And therefore suddenly this isn't quite so surprising anymore because there's a very 389 00:39:34,780 --> 00:39:40,300 obvious change in the group theory structure between degree four and degree five, 390 00:39:40,300 --> 00:39:43,780 which you don't see if you just look at polynomials themselves. 391 00:39:43,780 --> 00:39:50,270 And this is in some kind of advert for why abstract mathematics is actually has a point to it. 392 00:39:50,270 --> 00:39:55,060 It allows you to better solve the original problem that you were trying to. I couldn't agree more. 393 00:39:55,060 --> 00:40:01,510 In the final minutes of the show, we've mentioned Chevallier who took out US papers and published them initially, 394 00:40:01,510 --> 00:40:03,340 ten years later gave them to Louisville, 395 00:40:03,340 --> 00:40:11,770 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. 396 00:40:11,770 --> 00:40:18,700 What would you say are the most important applications of Gallas work and maybe more importantly, his ideas? 397 00:40:18,700 --> 00:40:25,030 Tsoukalas work has been incredibly influential across abstract algebra and just 398 00:40:25,030 --> 00:40:29,860 in terms of the sheer number of questions that this gives you more access to. 399 00:40:29,860 --> 00:40:35,530 So, for example, number three, my area of research and in fact all of our possessions, 400 00:40:35,530 --> 00:40:40,150 one big area of number three is all about can you solve certain equations? 401 00:40:40,150 --> 00:40:45,070 So not necessarily just polynomial equations, but maybe equations in two variables. 402 00:40:45,070 --> 00:40:53,080 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 403 00:40:53,080 --> 00:41:00,940 two variables and understanding these kind of things from different viewpoints, 404 00:41:00,940 --> 00:41:06,610 which is what Gaulois gives us access to. If we look at absolute algebra is just incredibly, incredibly useful. 405 00:41:06,610 --> 00:41:10,600 But outside number theory, I mean, has abstract algebra had applications in other areas? 406 00:41:10,600 --> 00:41:14,380 Yeah, I mean, it's an absolute algebra has just become its own subject. 407 00:41:14,380 --> 00:41:19,690 I mean, you know, there are plenty, you know, the whole research on here, which does, you know, algebra or representation theory, 408 00:41:19,690 --> 00:41:26,110 which are both offshoots of maybe the stuff that Gaulois started researching all those hundreds of years ago. 409 00:41:26,110 --> 00:41:34,210 Almost all areas of pure mathematics now involve some form of abstract algebra and not just pure mathematics, applied mathematics as well. 410 00:41:34,210 --> 00:41:38,500 And that's why we talked a lot about how when First-Year mathematicians come here. 411 00:41:38,500 --> 00:41:42,580 What group theory, though, that there's a reason why you learn group theory in first year, 412 00:41:42,580 --> 00:41:48,550 and it's because it underpins all of us so much the mathematics that you do later on. 413 00:41:48,550 --> 00:41:54,190 So pretty much any area of mathematics that we could mention will have kind of important one. 414 00:41:54,190 --> 00:42:03,190 In particular, any theoretical physicist, you know, a huge amount of group theory because it underpins all the chemistry that chemistry, 415 00:42:03,190 --> 00:42:08,290 these permutations or symmetries are also important in studying the structures. 416 00:42:08,290 --> 00:42:14,440 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. 417 00:42:14,440 --> 00:42:26,188 And thank you for listening.