I took this in grad school decades ago, and was confused about how Galois, or anyone, could have come up with it, which, of course, the way it was taught, they couldn't have (since it was backed into, knowing things that weren't known until later.) For years I've wanted to know Galois' thinking, so it was great to see this. It makes a lot more sense looking at it in its historical context.
on a high because your wizardry just puts it all into place! wow, just superb lectures. there's much to digest in Group/Galois theory but appetite is absolutely whetted - thank you Mr Wildberger, thank you!
Key point for me (starting ~@3:40): Galois considered fields extended by rational functions in the variable p, and that enabled him to make the correspondence between permutation groups and fields.
Excellent presentation. I followed a course in Galois theory at university - and the fascination stuck with me This reduction of degrees by finding symmetries is very reminiscent to the way you reduce the the order of an ODE and number of independent variables in a PDE by a applying the symmetry groups of transformation for the equations. There is definitely a connection somewhere. This must be the first math lecturer I have seen that writes entire sentences on the blackboard :-)
Wow - thank you so much for your awesome videos/lectures. I have really been enjoying them and they have been a great help in my learning. Thoemas, Copenhagen.
Awesome. Professor, do you have any donation page? If so, you should advertise it in your video; it doesn't have to be the "in your face" kind, but you should have one, because what you're doing is great and I'm sure a lot of people would be ready to help you expand the horizons (e.g. get a bigger viewership, be able/free to produce more content, etc.).
Thank you very much. That’s given me a glimpse of what Galois was actually doing and what solving polynomial equations has to do with group theory. Some analyst 😉
Always a pleasure to watch a master teacher at work. Thank you. But I noticed your crusade against infinity emerged at the very end. I have not understood this so far and I have been viewing your videos for years now. Also, I noticed on your web site it said only "Professor". 'bout damn time.
The crusade against infinity is ill-conceived. Wilberger wants to restrict the definition of "what is math?" down to what is finitely computable. But that's just a definition. It is fine. But other's who consider infinities part of legitimate mathematical analysis are not wrong, they just are doing something different to finite constructible math. The contention of the finitists is that there are logical flaws in any "mathematics" that include infinities. But they've never been able to show any explicit contradictions. The problems seem to be only conceptual, but that is no reason to reject transfinite analysis as "not mathematics". To claim transfinite number theory is not proper mathematics one needs explicit proof of contradictions, and that has never been forthcoming. In fact, if you dig into model theory, there is good reason to suppose that for some systems like ZFC no contradictions will ever be found, and so transfinite number theory can be considered legitimate mathematics. This does not mean restrictiing your own mathematical explorations to strict finitism is a bad thing, it is a good thing, as Wildberger notes, because computers cannot do infinite math, so if you want to work on rigorous computer math, you have to avoid hyperreals and other types of continua. It is a very good discipline to do that, but it does not at all mean that transfinite mathematics is wrong. An inexact metaphor would be that finitists like Wildberger are like engineers who want to stick to building land-based vehicles and refuse to even consider the possibility of flight. It is perfectly fine to limit oneself like this, it is a way of getting innovation. My view is just that the finitists should not be claiming their restricted territory is the only thing that qualifies as "mathematics". They do not own the definition of "math".
I fight side by side you Norman in your Algebraic crusade knowing as you do that the borders of algebraic kingdom have not be fully clarified yet, and abusively surunded to "continuous farytail". Nevertheless, dispite this clear consciousness, I will predict that this "Discrete" quest will fail at a crucial point which is ectremely worth seeking. Why? Because Discrete/Continuous is an archetype mater that conscious mathematics only touch half of it. It is this famous Quadratura del Circulo. Neighther archetype is reducible to the other. You will never Kill the Continous Dragon, nor reduce it to the Discrete one. They form a unitary and complementary undestructible couple. This is the true inconscient resistance that your marvelous work and quest is actualy facing in so many professional mathematicians. They may put in front lots of arguments, little of which are truely robust, but their actual resistance is mainly inconscious and takes its roots in the Quadratur del Circulo metaphysical central principal. You will surely resist yourself to this point of view, since you still find marvelous victories on falsly Continous lands that you discover belongs to Discrete kingdom. But you have not the Big picture of the Math Suprem Puzzle. And since you rely more on fulgurant intuition that you officialy admit, I predict that you will accept at one point to open and hear this complementary point of view. To give one glance out of many, shroedinger différential équation is a very hard problem, alique 5th power polynomial equations, because you canot factorise it via "algebraic" operator procedure. It leads to a riccati which is as hard as the starting shroedinger. And it's almost all what is usually done about it, on an algebraic point of view, letting it entirely in the hand of numerical analysis to make mince meat out of it. Nevertheless there is an other path, but that demand to abandon a sacro saint metaphysical or psychological Axiom : the belief in EQUALITY supremacy. Indeed as soon as one accept to loose the extreme rigid prison of EQUALITY, seeking no more exact solutions but EQUIVALENT CLASSES of solutions, then the Lagrangian perturbation theory can be applied in a breaking through way, that allows to CRACK THE PROBLEM. The pertubative parameter epsilon, playing the role of a potential infinitesimal, that link the two borders of the river : Discrete and Continuous paradigm. In other words the limit of your inlightening crusade is traped since the very begining in the jail of rigid EQUALITY IDEOLOGY, apparently unquestioned. Perhaps taking into acount this big potential picture may even bring some insight keys to the frontal crusade and one way quest. It may be like diging à path under the Wall. Best regards
This is definitely one of the best videos on the topic, thank you.
Best explanation of Galois correspondence I have seen. This video is a treasure. congrats
Thank you!
I took this in grad school decades ago, and was confused about how Galois, or anyone, could have come up with it, which, of course, the way it was taught, they couldn't have (since it was backed into, knowing things that weren't known until later.) For years I've wanted to know Galois' thinking, so it was great to see this. It makes a lot more sense looking at it in its historical context.
on a high because your wizardry just puts it all into place! wow, just superb lectures. there's much to digest in Group/Galois theory but appetite is absolutely whetted - thank you Mr Wildberger, thank you!
Key point for me (starting ~@3:40): Galois considered fields extended by rational functions in the variable p,
and that enabled him to make the correspondence between permutation groups and fields.
Your educational qualities are amazing.
The math does not feel complex when you get a historical perspective.
Thanks for everything.
Excellent presentation. I followed a course in Galois theory at university - and the fascination stuck with me
This reduction of degrees by finding symmetries is very reminiscent to the way you reduce the the order of an ODE and number of independent variables in a PDE by a applying the symmetry groups of transformation for the equations. There is definitely a connection somewhere.
This must be the first math lecturer I have seen that writes entire sentences on the blackboard :-)
Thank you Professor for giving it to us. It was a pleasure watching and learning a lot from you.
Thank you, Galois nailed it - one of my heroes
He nailed it before getting nailed the following morning. He was consistent.
Wow - thank you so much for your awesome videos/lectures. I have really been enjoying them and they have been a great help in my learning. Thoemas, Copenhagen.
+Thoemas You are welcome. If you like you can join the Patrons of my channel, at www.patreon.com/njwildberger?ty=h.
I was finding such stuff very long... i addicted with algebra its one of my favourite topic
Thank you for these series of lectures. I enjoyed them a lot. You are very clear and prompt. Cheers
You're welcome.
AWESOME. The example to explain how Galois theory works is so clear and intuitive. Beautiful!!!
Thanks!
Alex Gong But you know that it is just “the big picture“, right? To really understand that stuff you’ll have to dive into details.
This is great and clear. Galois theory in one lecture. Wow
Awesome.
Professor, do you have any donation page? If so, you should advertise it in your video; it doesn't have to be the "in your face" kind, but you should have one, because what you're doing is great and I'm sure a lot of people would be ready to help you expand the horizons (e.g. get a bigger viewership, be able/free to produce more content, etc.).
Thanks, I do now have a Patreon page at www.patreon.com/njwildberger
Thank you very much. That’s given me a glimpse of what Galois was actually doing and what solving polynomial equations has to do with group theory. Some analyst 😉
Always a pleasure to watch a master teacher at work. Thank you. But I noticed your crusade against infinity emerged at the very end. I have not understood this so far and I have been viewing your videos for years now. Also, I noticed on your web site it said only "Professor". 'bout damn time.
The crusade against infinity is ill-conceived. Wilberger wants to restrict the definition of "what is math?" down to what is finitely computable. But that's just a definition. It is fine. But other's who consider infinities part of legitimate mathematical analysis are not wrong, they just are doing something different to finite constructible math. The contention of the finitists is that there are logical flaws in any "mathematics" that include infinities. But they've never been able to show any explicit contradictions. The problems seem to be only conceptual, but that is no reason to reject transfinite analysis as "not mathematics". To claim transfinite number theory is not proper mathematics one needs explicit proof of contradictions, and that has never been forthcoming. In fact, if you dig into model theory, there is good reason to suppose that for some systems like ZFC no contradictions will ever be found, and so transfinite number theory can be considered legitimate mathematics.
This does not mean restrictiing your own mathematical explorations to strict finitism is a bad thing, it is a good thing, as Wildberger notes, because computers cannot do infinite math, so if you want to work on rigorous computer math, you have to avoid hyperreals and other types of continua. It is a very good discipline to do that, but it does not at all mean that transfinite mathematics is wrong.
An inexact metaphor would be that finitists like Wildberger are like engineers who want to stick to building land-based vehicles and refuse to even consider the possibility of flight. It is perfectly fine to limit oneself like this, it is a way of getting innovation. My view is just that the finitists should not be claiming their restricted territory is the only thing that qualifies as "mathematics". They do not own the definition of "math".
Thank you for making some sense of Galios theory.
Great lecture! Thank you very much.
I was trying to understand Galios theory by myself, but was stuck. This is incredible.
Thanks Arif, I am glad it was helpful.
thank you so much for your wonderful videos !!!!!
Glad you like them
good lecture
Fantastic!
This is awesome. Thanks!
fantastic, thank you!
You're welcome!
I fight side by side you Norman in your Algebraic crusade knowing as you do that the borders of algebraic kingdom have not be fully clarified yet, and abusively surunded to "continuous farytail". Nevertheless, dispite this clear consciousness, I will predict that this "Discrete" quest will fail at a crucial point which is ectremely worth seeking. Why? Because Discrete/Continuous is an archetype mater that conscious mathematics only touch half of it. It is this famous Quadratura del Circulo. Neighther archetype is reducible to the other. You will never Kill the Continous Dragon, nor reduce it to the Discrete one. They form a unitary and complementary undestructible couple. This is the true inconscient resistance that your marvelous work and quest is actualy facing in so many professional mathematicians. They may put in front lots of arguments, little of which are truely robust, but their actual resistance is mainly inconscious and takes its roots in the Quadratur del Circulo metaphysical central principal. You will surely resist yourself to this point of view, since you still find marvelous victories on falsly Continous lands that you discover belongs to Discrete kingdom. But you have not the Big picture of the Math Suprem Puzzle. And since you rely more on fulgurant intuition that you officialy admit, I predict that you will accept at one point to open and hear this complementary point of view. To give one glance out of many, shroedinger différential équation is a very hard problem, alique 5th power polynomial equations, because you canot factorise it via "algebraic" operator procedure. It leads to a riccati which is as hard as the starting shroedinger. And it's almost all what is usually done about it, on an algebraic point of view, letting it entirely in the hand of numerical analysis to make mince meat out of it. Nevertheless there is an other path, but that demand to abandon a sacro saint metaphysical or psychological Axiom : the belief in EQUALITY supremacy. Indeed as soon as one accept to loose the extreme rigid prison of EQUALITY, seeking no more exact solutions but EQUIVALENT CLASSES of solutions, then the Lagrangian perturbation theory can be applied in a breaking through way, that allows to CRACK THE PROBLEM. The pertubative parameter epsilon, playing the role of a potential infinitesimal, that link the two borders of the river : Discrete and Continuous paradigm. In other words the limit of your inlightening crusade is traped since the very begining in the jail of rigid EQUALITY IDEOLOGY, apparently unquestioned. Perhaps taking into acount this big potential picture may even bring some insight keys to the frontal crusade and one way quest. It may be like diging à path under the Wall. Best regards