Why you can't solve quintic equations (Galois theory approach)
Vložit
- čas přidán 16. 05. 2024
- An entry to #SoME2. It is a famous theorem (called Abel-Ruffini theorem) that there is no quintic formula, or quintic equations are not solvable; but very likely you are not told the exact reason why. Here is how traditionally we knew that such a formula cannot exist, using Galois theory.
Correction: At 08:09, I forgot to put ellipsis in between.
Video chapters:
00:00 Introduction
00:23 Chapter 1: The setup
04:38 Chapter 2: Galois group
11:15 Chapter 3: Cyclotomic and Kummer extensions
19:43 Chapter 4: Tower of extensions
27:25 Chapter 5: Back to solving equations
35:23 Chapter 6: The final stretch (intuition)
43:25 Chapter 7: What have we done?
Notes:
I HAVE to simplify and not give every technical detail. This is made with the intent that everyone, regardless of their background in algebra, can take away the core message of the video. This can only be done if I cut out the parts that are not necessary for this purpose. As with my previous video series on “Average distance in a unit disc”, this is made to address the question I always had when I was small - treat this as a kind of a video message to my past self.
For the “making everything Galois extension” bit, we will need to show that the only things fixed by ALL automorphisms over Q must be in Q itself. This is intuitive, but difficult to justify rigorously. All proofs I know involve “degree of field extension”, and the very satisfying result called the “tower law”, which I deliberately avoided throughout the video because it turns out not to be necessary for the core part of the video. For instance, this proof: math.stackexchange.com/questi...
The reason we have this mess is that we defined Galois extension using the splitting field of a (separable, i.e. no repeated roots) polynomial. The usual definition given is exactly as above - only things fixed by ALL automorphisms over Q must be in Q itself. This typical definition will of course solve the problem above, but will now create the problem of why this definition implies the larger field is made by adjoining the roots of some polynomial. These two definitions are equivalent, but I just think that it makes much more sense to define it the way I did in the video, in the context of the video; and also I think this is an easier definition to accept.
Resources on Tower law: en.wikipedia.org/wiki/Degree_...
artofproblemsolving.com/wiki/...
Quotients of solvable groups are solvable (the elementary proof): math.stackexchange.com/questi...
[The question is already the proof - it is a really elementary way to show the result that we want]
More resources on proofs that A_n is not solvable:
Sign of permutations: en.wikipedia.org/wiki/Parity_...
Alternating groups: mathworld.wolfram.com/Alterna...
The proof that A_n is simple (i.e. no non-trivial normal subgroups): ramanujan.math.trinity.edu/rda...
[You need to only go up to Page 5 towards the end of the proof of Theorem 2, but you definitely need group theory lingo]
If you know a bit of group theory (orbit-stabiliser and Cauchy), then you can see that the polynomial x^5 - 6x + 3 has the full S5 Galois group, because it is (i) irreducible [this requires Eisenstein’s criterion, see link below], and (ii) exactly two complex roots [and hence the Galois group contains a transposition, i.e. complex conjugation]. Note that the Galois group is transitive. This again needs quite a bit of justification. For the proof assuming transitivity, see here: math.stackexchange.com/questi...
Eisenstein’s criterion: en.wikipedia.org/wiki/Eisenst...
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
Facebook: / mathemaniacyt
Instagram: / _mathemaniac_
Twitter: / mathemaniacyt
Patreon: / mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
Ko-fi: ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
Correction: 36:31 S_5 is the symmetric group of degree 5, not order 5.
Edit: This is now an entry to #SoME2. While this channel is not big in the grand scheme of CZcams, it is still much more established than the intended participants, so I am not expecting this to win whatsoever, and I treat this as a "symbolic" submission. Hopefully such a hashtag could act as a signal boost to the competition, precisely because this channel is more established. More details on the community post: czcams.com/users/postUgkxizni2JAp8Nw-xOvPI1ut2aNcRhXfMUzQ
As a result, during the peer review process, if I am your video's intended audience, then I would do my best to give constructive comments on the video. I highly encourage viewers like you to check out all the videos with the #SoME2 tag, and give all those video creators some love.
By far the most ambitious video - it took quite a bit of time, and lots of effort went into it! Please do consider subscribing, commenting and sharing the video. And if you want to support the channel, consider going to Patreon: www.patreon.com/mathemaniac
“If only you knew the magnificent of 369… you will have the key to the universe” - Nikola Tesla
I had to pause the video after 5 seconds and look up what a "quintic equation" is. A quick statement describing this a polynomial of degree 5 would help the casual viewer from being lost in the opening sequence.
The truth is that there are many words by which a person can express his admiration for something, but for us there is a word that we use frequently when the astonishment is beyond description, and it is (subhan Allah), and it can be translated in English to (Glory be to God), but when it is pronounced in Arabic, the astonishment is double. .
I paused the video at 14 minutes to write this comment. I am amazed at the beauty of your explanation and the genius in your simplification of those terms that no university professor could simplify in this way.
Thank you from the bottom of my heart.
@@flippy3984
The universe is magnificent, but Nikola Tesla was a fool and a con man.
369 is a number. Bleahhh.
Another correction, at 8:18 you missed the "..."
Evariste Galois was so impressive.. especially considering the fact that he invented all this before he died at 20 years old
"Evariste Galois was so impressive.. especially considering the fact that he invented all this before he died at 20 years old!"
How many people have invented something after they have died? 😀
@@keinKlarname lol the emphasis isn’t on the fact that he invented something before he died.. but rather his age when he died. The comment would be basically unchanged in meaning if I dropped “he died”
@@keinKlarname Nice play with words... Liked
I wonder how many others like Evariste Galois there are who don’t get a chance to invent something no matter what age they live to.
@@thomaskember3412 oh 100% agree
Your presentation is always clear and relaxing! I have to spend more time later in order to understand this video. Great work as always!
17th liker moment22hrslate
Nice!
I had a bit of trouble following it the first time, to say the least. I am glad I am apparently in good company!
If one needs to spent more time later, maybe it means the content is not clear enough. Try this one and compare: czcams.com/video/CwvuZ8aHyH4/video.html
thank you Mathemaniac and blackpenredpen for all your amazing videos ❤
Brilliant exposition.
And, as it happens when it's about real mathematics, only a few numbers were mentioned: 1,2,3,4,5.
and 23
@@blockmath_204823 has a 2 and a 3 /s
When I was learning this I couldn't really figure out how to explain this to anyone who hadn't done group theory yet, which is a sad fate for a subject as beautiful as Galois theory. Kudos to you for explaining it so well!
I definitely have to invest more time until I can thoroughly digest all of this information. But I think this video has already helped me immensely in my quest to get there. Taking a step back, it seems incredible that this is available for free on CZcams. Thank you so much for taking the time to make such top-notch material, and I hope that you keep up this amazing work for the foreseeable future!
Thank you for the kind words!
@@mathemaniac why does math TAKE SO.DAMN LONG FUCK MAN IM TIRED..whatif I DONT WANT TO WATCJBTHIS MORE TJAN ONCE TONGET IT..I want to be a math genius and not have it take so damn long ti learn all this..surely there must be a way to learn faster??
@@leif1075 speed run math ig?
@@leif1075 Why? The process *is* the math. You aren't getting smarter by watching youtube videos, you are just getting more informed. More than that, if you want to have even a chance of really understanding this, you'd have to spend at least another week studying this, it's corollaries, and any related mathematics, as well as actually be working out some problems with a pen and paper. You think you can watch one video on something and now you're an expert? LOL.
r-
[The pinned comment got removed by CZcams, again...] This is a very ambitious video, and it took me a lot of time and effort - please like, subscribe, comment, and share this video! If you can, please support the channel on Patreon: www.patreon.com/mathemaniac
A bit of remark:
I HAVE to simplify and not give every detail. The intent of this video is to not dumb it down too much, but at the same time not give every technical detail so that it is still accessible. The final bit of (a) why S_5 is not solvable, and (b) why any particular polynomial has Galois group S_5, are dealt with by intuition, and I do expect people to come unsatisfied with this. However, I still leave out those details because it uses more group theory than I would like to include in the video (actually it is also because I have a bit of crisis making such a long video).
For (a) in particular, if you know group theory and the proof, I hope you agree that group theory is only slightly more civilised than "brute force" - essentially those constraints allow you to brute force everything, but group theory allows you to skip quite a bit of calculations, but it still leaves you with quite a few cases you need to deal with. In fact, I have actually flashed out the sketch of the proof on the screen. For people who don't know group theory, it will feel as though somehow magically things work out in S_5, but it does not answer the "why".
For (b), it starts with theorems in group theory (and ring theory) to get you started, but ultimately it is still a bit of fiddling things around and again magically the Galois group is S_5. So again, it would not answer the "why", and so I appealed to intuition saying that most quintics are not solvable.
As said in the video, if you want the details, go to the links in the description; but honestly the best approach would still be studying group theory in more detail.
But in any case, I do hope that you are motivated to study group theory because of this - but I have to be honest, don't study Galois theory JUST because you want to know this proof in more detail. Galois theory is difficult, and it is actually pretty ridiculous and ambitious for me to even attempt to make this video. Study Galois theory only if you are really into abstract algebra and like playing around with these abstractions.
Any idea why YT removed a pinned comment? Bug in the comment system? Trigger words?
@@MichaelPohoreski I have asked CZcams about it, and at first they insisted that they couldn't do anything about it, but then I also insisted that it was actually CZcams who deleted it, so there must be a way to restore the pinned comment. It turned out that CZcams mistakenly removed the pinned comment because it violates CZcams policies (which it clearly doesn't), and claimed that "mistakes happen", even though most of my pinned comments in the past half of the year got removed by CZcams as well. I really hope that they resolve this issue soon - it has been more than half a year since they did that to my pinned comments.
@@mathemaniac That has to frustrating as hell that YT is _that_ incompetent and can’t even track down what caused the problem in the first place!
I guess keeping a history with a “reason” would be too much “work”. /s
Least you got your comment back. YT is getting more and more false positives with smaller channels having no recourse except to hope that a bigger channel picks up on it.
This quality of content is crazy!
I just finished my 2 semesters of group theory and ring theory, so thanks a lot for giving me the next part!
Good, now you can stand in unemployment line. You can think of galois groups while flipping burgers.
@@reimannx33 good. Now you can stand in the "garbage person" line. You will think of this when being exposed in the middle of the street. You probably have a low paying job
@@reimannx33 Math is actually a pretty good major lol
Thank you for the time and effort put on the video. Channels like yours make the world a better and more interesting place to live on.
Fascinating topic and a beatiful example of the abstration capabilities of human reason.
Abstract math is difficult for me. I appreciate high quality videos such as yours to help mitigate that struggle :)
You made this about as clear as possible without a full course. I have definitely gained insight into this difficult area.
I hope this channel gets more views. The editing and audio quality are fantastic
Wow! This was such a great video. I've been wanting to see something like this for a long time. I much better understand the unsolvability of the quintic now! You've gained a subscriber!
Love this video, I wish it existed sooner! I spend the last semester working up to this proof and rigorously proving all the little steps you had to skip over, and it would have helped to go in knowing the big picture. This video would have given that big picture.
For a man who was alive for such a short time, Galois truly did manage to live forever 🙏🙏
Great video! I never quite understood this topic, but thanks to your video, I gained a much better intuition about it. :) Thanks for making this video and putting so much effort into it!
Beautifully done! Cannot even begin to imagine how much time and work went into this. Manim's difficult to use but you did it incredibly well!
Thanks for the kind words! But as I stated in the description, I don't use Manim actually.
@@mathemaniac Ohh I see, my bad! Even more impressive- I'm in a similar boat where I'm using less manim each video, so it's cool seeing another channel doing the same!
I've been looking for a video like this for a long time. You're the next 3blue1brown!
An excellent video on a topic I have been wanting to learn more about. Thank you so much!
Although there are some things that could be improved, the video is really good and informative. The main concepts are there and I appreciate what you've done. It's given me the confidence to look more into the topic. Thanksb
This was amazing, thanks a ton! To summarize:
- The process of solving equations via +-*/ & radicals is equivalent to starting with a base field of accessible elements, & then including new layers of numbers [which are roots of the currently accessible field elements].
- This [cyclotomic+Kummer] extension tower has a very specific property, that the symmetries of the newly included numbers over the previous layer always contain the previous symmetries as commutative-normal-subgroups.
- Alternating group A5 has no non-trivial normal-subgroups, it's the smallest non-commutative simple group. We run into this when looking at S5 symmetry of some quintic equations.
- The previous points imply that extending layers of radical expressions of field elements can never reach quintic structures.
Please correct me if I'm missing anything here.
This sounds very similar to a high level sketch for proving which numbers are constructible [only the ones which we can reach through tower of field extensions of degree 1 or 2]. There is an arxiv which also has a beautiful representation of A5, Galois Theory : A First Course - which, as the author explains, coincidentally looks like the simplex known form of Carbon :)
Good addition to 'Aleph 0's and 'not all wrong's videos. This topic definitely needs more exposure and expositions!
one of the ( if not the ) best channels of maths out there
i love that you deal with topics which are at a higher level than most of math content on youtube
Thanks so much for the kind words!
3B1B. I’m guessing you’ve never heard of that channel.
@@ophello to be honest I like this one more, even though 3b1b created the tool used for these videos, mathemaniacs videos are more useful and deal with higher level topics. yeah 3b1b channel id great but my 2 favorites are this one and oljen ( french channel )
I'm still listening and looking at the video and I'm already thinking this is the best explanation of Galois theory I've ever found! (and I really spent a lot of time in the past about this topic)
This is SO hard but I've been so curious about it for ages, props on the video!
What is thought provoking is 6 is being shorted in the whole of it.
Thank you for uploading this! Can't wait for more cool long form content!
Congratulations! This is a masterpiece! Thank you so much
Just discovered this channel! 🎉 I’m delighted at seeing it’s high level math. 🙌
This is brilliant. Beautiful. Pure delight.
This video is basically a cogent recapitulation of the second half of second term graduate school algebra. Quite impressive.
In Romania this is done in the third year second semester undergraduate.
Outstanding as always, with each video you keep setting the bar higher!
Glad you like them!
this work is phenomenal ! congratulations
Abstract algebra has always been a weak point for me. However, since I am interested in algebraic topology, I figured I should get comfortable with computing the homology and cohomology groups. My ultimate goal is to study the application of Lie groups to differential equations. I was inspired by the Frobenius Thm, and it just sort of clicked.
Picking up abstract algebra again, I find I am engaging with the material in a new way. Exact sequences led me to study normal subgroups and quotient groups. But now, I can also see the significance of permutation groups. Your video gave me a quantum leap into the Galois theory endpoint. Galois groups motivate the prerequisite material very nicely.
Stop hiding from having to get a job. I am sure frobenius will help flipping burgers.
@@reimannx33 Do laws and set principles on any subject make it true?
Thank you for doing such a brilliant video on this topic! I am a uni student struggling with this course but your exposition makes it 1000 times clearer than all I had learnt earlier!! Thank you!!
Explaining quotients with buckets is genius, thank you for the idea
I'm so glad I found this video. This is amazing.
I'm going to have fun with this one, thank you so much for your efforts.
Also I feel like permutations are a good way to describe automorphisms in an accessible way
Very very thorough and interesting. Such a shame you don't have more views!
Fantastic content dude, thanks this is very helpful
Awesome . Understanding this deeply is one of the things i want to do before I die ☺️. Nice video!
Galois Theory is such an interesting branch of mathematics and you did an awesome job explaining it! :)
I love seeing stuff like this on CZcams. A lot of people think computers are the be all end all of mathematical processing - and that’s true, to a point. Computers are phenomenal at simple operations, and sorting. Computers are not any kind of good at abstract mathematics. They’re slowly getting better, but they don’t have intuition, and they aren’t able to substitute or generalize well.
People who can do complex and abstract math are never in huge numbers, but are badly needed for scientific advancement.
Keep being awesome.
Cheers!
Fantastic video. Thank you so much!
I was very impressed! Amazing video
omg aaaah sooo excited. I've been really interested in really understanding this.
Honestly, this video is also a message to my past self who is very eager to know about this problem - so you are not alone in this!
Thank you for this video!
Awesome, sir. You are my favourite Teacher. Keep doing...
Love from Bangladesh
❤️
this is so great, thank you! finally things are clearer to me!
Brilliant explanation!
Totally amazing presentation. Thanks so much. Reading Fields and Galois Theory by John Howie.
I found out about #some2 today im really glad that this exisits. Many new channels start like this and I discovered few really good channels like yours
I have no business being here but I understood a scary amount of this my first watch through, probably just a tiny fraction of the whole. You do a very good job explaining and speaking clearly, thank you
I am not highly skilled in the theory either, but I know the star pattern is a can of worms.
Thank you, this will need a bit of coming back!
Honestly it will be impressive if you get everything in one viewing - even 45 minutes is very concentrated for this really difficult subject of Galois theory.
Wonderful video!!! Thank you so much and please know that we will always support you
I really appreciate the effort here -- can't feel bad that I didn't come out with a complete understanding.
Got some more reading to do.
Thanks for the appreciation! I don't expect anyone to come out with a complete understanding, and actually did say in the video that it may require a second, maybe third (and maybe more) viewing in order to really understand!
Simply wonderful.
Simply amazing. I loved this video. Your explanation is quite neat and intuitive. You made me understand Galois theory in less than an hour! Great job! you have a new subscription :D
P.S thanks for all the work you put into this video
Glad you enjoyed it!
Excellent explanation and presentation, better than others.
VERY GOOD. Amazing video with such interesting content.
This is good! I understood the normality of the automorphisms nicely!
really good job - well done - thank you sir
That was really overwhelming. I'm not very good at group theory yet, so everything was somewhat new to me. However I really want to understand these concepts. I think I'm watching it again. Anyway, thanks for the video and the hard work with it!
Basic group theory in and of itself is easier to understand than Galois theory. This however is the application that made Galois invent group theory. It is outrageous that mathematicians hadn't (in modern history, post 9564 BC i.e.) already invented group theory for the purpose of studying symmetries as such, rather than for studying polynomials and fields.
@@henrikljungstrand2036that is operating on hindsight in my opinion. For most of history, anytime an argument relying on symmetry is needed, you would write in prose and it would be understandable. Studying symmetry for it's own sake was done with plane tilings, combinatorial problems, etc, but it's not obvious unifying those things is fruitful. It would seem unnecessarily algebraic to use letters to play with symmetries of a shape/objects (even negative numbers were strange to Europeans).
And there is bad timing: accepting negative numbers and abstraction comes with the advent of calculus and the scientific revolution. Those are quite some rabbit holes, and F=ma just feels like a better direction than e=a*a^(-1).
@@hybmnzz2658 You have some good points here, but i still think we should historically have had a mathematical theory of the transformative concrete symmetries of geometrical objects, like e.g. rotations of the regular icosahedron, that leaves it in the same overall appearance as before.
A transformative symmetry is a symmetry you perform as a motion or change, but which "surprisingly" leave you in an equivalent state to the one you started with. It is a special case of a transformation, that usually changes the appearance.
A concrete symmetry is a symmetry that is part of the physical symmetries of space (or of "space time"), and which you may observe more or less directly.
This is the best explanation of why the Galois correspondence implies quintic insolvability that I've seen.
This was a great video, the only feedback I can give as a viewer who is new to the subject us that I would have liked a final concrete example. Just taking a single quintic and showing that it in particular is not solvable.
I did think about this: actually giving the proof of why x^5 - 6x + 3 has Galois group S_5, and hence not solvable; but the proof involves orbit-stabiliser theorem and Cauchy's theorem, which are too difficult for this video, if I am intending for an audience without much knowledge of algebra like yourself (maybe need an extra 10 minutes or so). This polynomial having Galois group S_5 is already the easiest to justify compared with the vast majority of others...
This is why I put links in the description so that people who know group theory could delve into it (maybe even turning it into an exercise), but not giving the exact details in the video; and instead, I just appealed to intuition saying that the roots are not very algebraically related, so the vast majority of polynomials are not solvable (which is true actually).
@@mathemaniac Thanks for the explanation. As I’ve been learning more higher-level math, a common theme has been that some topics are way easier or harder to explain than I expected. Overall still a great video, and afaik a good intro to the topic. I plan to revisit this once I know more of the prerequisites.
Instant sub , great video!!!
I had never been so excited for a math video, one of my favorite topics with one of my favorite youtubers ❤
Wow, thank you!
Another deep and awesome math video.
I just need to pass my Galois theory exam to get my masters and this video is brilliant helps so much with explaining the basics in a simple manner thank you 😁
I have kinda stopped watching maths videos
But thus video re-sparked my interest
Nice pentagram.
37(n+n+n)=nnn for 0 < n < 10. Sin (666) + Cos (6*6*6) = -1.618... (golden ratio). Belphegor's prime is the palindromic prime number 1000000000000066600000000000001 (10^30 + 666 × 10^14 + 1).
Awsome presentation thanks Mathemaniac
3:29 splitting field definition.
4:57 automorphism = a function from a field to itself fulfilling some properties. 1) maintain algebraic relation 2) identity mapping for a smaller field from which the larger field is extended 3) one-one mapping
Well worth your effort, I can assure you.
Amazing video!
will you do a video on differential galois theory? it's much more interesting imo
It definitely sounds interesting, because it is using algebra on analysis problems, but I have absolutely no knowledge of differential Galois theory at all. But I will look into it.
Have you learned any algebraic number theory? Galois theory plays one of the most fundamental roles, and is related to some of the deepest mathematics there is: Iwasawa theory, Galois representations and the Langlands program, the etale topology, and the list goes on. Differential Galois theory even enters the picture in certain areas of arithmetic geometry.
@@theflaggeddragon9472 no I haven't gotten in the number theory rabbit hole yet, but I'd love to! I hear it's huge.
Brilliant as always, Trevor. Delightful to be able to get a window to some higher-level math concepts before learning it formally. I was actually taking some physical notes along with watching the video because I tend to get mathematical definitions confused upon first exposure.
One question I had was for the point around 32:49. Am I right in thinking that the groups being Abelian is not important for being solvable, but since we are always able to arrange a solvable problem in terms of Abelian extensions then it is a restriction we are allowed to place? In other words, you could come up with a sequence of extensions solving the polynomial which are not Abelian, but there is always a way to restate them in terms of a sequence of extensions which actually are Abelian?
The Galois group of the quintic polynomial might not be abelian (actually the Galois group of x^5 - 2 isn't), but you have to find a chain of abelian extension to prove it is solvable. In the case of x^5 - 2, we found the chain: Q - Q(zeta) - Q(zeta, 2^(1/5)). At each step the Galois group is abelian.
Most Excellent presentation I've ever seen on this difficult topic. PLEASE make a similar post on the relation between DIV GRAD at finite density charge sources and the relation of this to gravitational curvature for finite density mass distributions. For zero charge density DIV GRAD X=0, while for mass the mass on a rubber sheet model suggests negative (Gaussian) curvature in the surrounding vacuum, suggesting DIV g
I find the quartic formula fascinating. How did people came up with that monster? Could do you a lecture explaining who and how they discovered the formula?
I'm not sure about the full history & original motivation for the idea - but there is actually a very clean approach using a resolvant polynomial [hard to invent, but reasonable to digest] that might interest you.
You have to go back and forth. Cubes then roots and factors. 9=3^2
3+2+2^2=9
So it's like a breakdown of operations. We create then identify the complexity of ordered sets. The 3 has to stay if there is a visual identity to include at least one plank of a z axis as imaginary numbers for the arithmetic to math.
Very Interesting!
TREVOR has created a visual video of Abel Ruffini Theorem about unsolvable quintics. And all in 45 minutes. This is a monumental achievement.
I highly recommend taking two semesters of abstract algebra first. I watched this before and after. It makes WAY more sense.
Amazing video, made me understand this topic a lot more. Thank you. Galois was a talented mathematician, so sad that he decided to die over a girl in a duel with trained soldier.
17:30 The "star-shape" is also a rotation by 4pi/5 (2 l pi/n in general).
Great video as always!
I knew that it looks like a rotation, but I deliberately cut that from the script, because it is a rotation on the roots themselves, but it is not a rotation on the complex plane, because things like 1 stays unchanged simply because it is in Q. I just don't want any confusion and cut that from the script.
@@mathemaniac That makes sense. I'm wondering what the map looks like using the ways in your previous videos to visualize complex functions. :)
Nice video, but a computer has it right?
More channels like 3blue1brown are popping up lately. That's nice to see
@Mathemaniac, thanks very much for the great video. I have read some basic Galois theory in textbooks, but parts of your exposition were clearer to me than Gallian's or Fraleigh's. Your argument has convinced me that if an extension field is a radical, Galois extension of a base field, its Galois group will be solvable. What I would love to know is: does the implication go in the opposite direction? If a Galois group is solvable, does that mean that the corresponding extension field will be a radical extension? It seems that your argument is based on the fact that the Galois group of a cyclotomic or Kummer extension is always Abelian, but for the argument to work in the opposite direction, it would need to be true that an Abelian Galois group implies a cyclotomic or Kummer extension, and I don't (currently) see why that would be true.
The opposite direction is also true, except proving it, as you said, does require quite a bit more machinery.
You need the structure theorem (actually a solvable group is usually defined with the quotients being cyclic rather than abelian, but you can check these two definitions are equivalent by structure theorem), which is not an easy fact; and then you need to prove that a cyclic Galois group (Zn) for the extension L/K, and K having the nth root of unity, implies that L = K(an nth root), which is again not an easy fact. I believe that should be covered in any standard Galois theory textbooks.
I chose to only cover the direction I gave because that is the only direction I need for proving quintics are not solvable, but the other direction is also worth looking into.
@@mathemaniac Thanks for the informative answer! If I can ask one more question, do you know of any textbook on Galois theory which is very approachable for a student with a good foundation in basic group, ring, and field theory?
As a computer scientist I appreciate the use of Manim and LaTeX, but that's about the most intelligent thing I can say here :)
Thanks!
This is very interesting. I am very familiar with groups (and, by extension, rings and fields), but I've never studied Galois theory. I personally find this video hard to follow, and yet I'm still getting a lot out of it. I think that it would help me to have certain definitions left on screen for the whole of a chapter (for example, I found it difficult to keep in mind what a "splitting field" is. It would have been nice for something like "Splitting Field: A base field adjoin the roots of a polynomial." to be on screen)
Thanks for the suggestion. I can't edit this video, but I will consider this in the future videos.
There are some glitches. For example, at 40:03, the arrangement of the triangular symbols that express the normal subgroup relation implies that G_{i+1} is a normal subgroup of G_i, not that G_{i-1} is a normal subgroup of G_i. (If K_1, K_2, ... is the tower of Galois extensions ordered by field inclusion and with L at the top, then G_{i+1} = Gal(L/K_{i+1}) is a normal subgroup of G_i = Gal(L/K_i).) I think this glitch appears more than once.
As long as I live I will never truly 'get' Galois theory, I don't know why it's so hard to understand vs other complex mathematics. Incredible that he was so young when he came up with it
excellent
How did you produce these wonderful graphics?
Here's an analogy to sudoku that makes sense to me:
We have defined a set of rules: for analogy purposes, we can pretend that it's the sudoku rule set. No two numbers can repeat in the same row, column, or box, yada yada.
X^2, x^3, and x^4 equations are like starting positions in sudoku that logically lead to a solution.
However, x^5 and above equations are like sudoku starting positions that have no logical path to the solution. That's not to say that there is no solution, just that there is no way to arrive at it without guessing.
There is simply not enough information to solve the puzzle. Or perhaps, the information has stayed the same ever since x^2, but we've expanded the grid again and again so that our information is now no longer sufficient to solve the puzzle.
Amazing and quest leading
Crunch/rip freeze prolong isolate 0:25 poison ivy DTG 2:29
Correction at 24:09 if i'm understanding correctly, just any automorphism of M won't do, it doesn't generally have to keep elements of L in L. For example select any element m0 from M that doesn't belong to L. Then the automorphism f(x) = m0*x trivially won't keep L fixed: f(L) != L. In short, we can't speak generally, but for our case an automorphism of M/K can be restricted to Aut(L/K).
Amazing
I'm missing something at 12:50 and beyond: why does the automorphism " of course must keep 1 invariant"? It's in Q, so the automorphism keeps Q invariant like e.g. complex conjugation keeps R invariant, so I see an analogy. But why does the root 1 must stay invariant by by the automorphism?
Nice video btw :)
Very Nice