What is the square root of two? | The Fundamental Theorem of Galois Theory
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- Опубликовано: 8 июн 2024
- This video is an introduction to Galois Theory, which spells out a beautiful correspondence between fields and their symmetry groups.
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SOURCES and REFERENCES for Further Reading!
This video is a quick-and-dirty introduction to Galois theory. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.
(a) Galois Theory
Galois Theory notes by Tom Leinster: These notes are by far the best resource out there for learning the subject. They’re completely rigorous, but they’re also written in a very reader-friendly way with lots of examples and motivation. (See link here: www.maths.ed.ac.uk/~tl/gt/gt.pdf)
This playlist on Galois Theory by Professor Richard Borcherds is a gem. It explains Galois Theory from the ground up, rigorously, in almost complete generality. ( • Galois theory )
(b) Group Theory
Group Theory lectures: This playlist by Professor Benedict Gross is a beauty. It goes through the entire group theory syllabus from the ground up, and Professor Gross is a masterful lecturer. (see link here: • Abstract Algebra )
MUSIC CREDITS:
The song is called “Taking Flight”, by Vince Rubinetti.
www.vincentrubinetti.com/
THANK YOUs:
Extra special thanks to Davide Radaelli and Grant Sanderson for helpful conversations while making this video!
SOFTWARE USED:
Adobe Premiere Pro for Editing
Follow me!
Twitter: @00aleph00
Instagram: @00aleph00
Intro: (0:00)
What is the square root of 2?: (1:08)
Fields and Automorphisms: (6:04)
Examples: (8:55)
Group Theory: (16:34)
The Fundamental Theorem: (18:25)
You just made me, an analysis lover, watch 25 minutes of abstract algebra content. That is incredible
I used to classify myself as an "analysis person" and later on an "algebra person". But eventually I learned that the most beautiful math happens when algebra and analysis mix together. Some subjects that show off their interplay include algebraic number theory, Lie groups, Hodge theory, elliptic curves, modular forms, and much much more! The proof of Fermat's last theorem uses LOTS of analysis and algebra for example, and all the subjects I listed above.
exactly my thoughts.
Incredibly well done.
@@theflaggeddragon9472 Llevo años aprendiendo Math y cuando uno se atasca en algún momento, siempre hay personas maestras que saben enganchar a la Maravillosa Matemáticas
@@renatohugoviloriagonzalez4881 Que bien que continuas aprendiendo! Y tienes razon que somos muy afortunados teniendo estos videos de espertos gratis en RUclips, para aprender y para desatascarnos. Aproposito, has visto los videos de Profesor Richard Borcherds?
Disculpe mi Espanol, no he hablado en much rato.
This video is a thing of genuine beauty. You have a rare talent for illuminating these deeply technical subjects in a fascinating and accessible way. Many thanks.
Thanks for the kind words, Jim! Appreciate you watching :)
Thank you for putting so much effort into making this. This is my first time hearing about Galois Theory, and this video was amazingly clear and a treat to watch. It's sad that so few people watch these compared to other channels of equal quality.
Nah give it some time, these vids get a sizeable number of views
@@NonTwinBrothers yeah
@@NonTwinBrothers I agree. It's not one of these channels that live from immediate hype. I wouldn't wonder if this video is still getting watched in 20 years, when 99.99999% of youtube content is long lost to irrelevance :-)
@@harriehausenman8623 Exactly. No one knows/cares who the Kim Kardashian equivalent of the 18th century was, but we all know who Euler or Bernoulli was.
@@elidrissii 🤣
Took group theory and ring and field theory but didn't get all the way to Galois theory by the end. What a clear and concise way to encapsulate the fundamental concept. Thanks for this.
Me too and I found abstract algebra the most difficult of all maths subjects I took. Watching this I felt like I was getting an insight and then it disappeared again, my brain obviously cant work in these terms.
Took Abstract Algebra and did much of what you did, with groups, rings, fields and the various domains (PID, ID) and ended after only after hitting a couple of the special topics (we hit finite simple groups, but missed intro to Galois theory)
holy shit you made the how fast is melee video, greetings fellow melee and math nerd lmao
Is there a second part where he actually explains why the 5th degree equation cannot be solved ?
@@sambtt thanks fellow nerd! :)
Huge fan of your explanation style and visuals! Can't wait to watch this.
When you said that the lines wouldn't go where I expected, I almost paused the video, because I was pretty sure I did see where they would go -- and I was right! My intuition was based on the understanding of multiplication by any complex number of magnitude 1 as rotation -- which of course wraps around after each full turn. So, ζ⟶ζ² applied twice is just the rotation by ζ² twice -- ζ² * ζ² = ζ⁴. Well, that's reasonably obvious, but the next step falls out of the wrapping nature of the rotation. ζ⁴ twice is just ζ⁸, but since ζ is the 5th root of unity, that means that ζ⁸ = ζ³ -- and so on. With this view, all of the graphs are immediately evident from the starting point of the given mapping of ζ.
Same thing for me, except I was thinking in terms of modular arithmetic.
I find it difficult to express just how GOOD this video was at explaining the general idea behind Galois theory. Genuinely, thank you. Thank you so much, you've given me another way to look at fields. Another tool that I didn't know even existed.
It's a testament to the complexity of groups and Galois Theory that simplified explanations still manage to fly over your head, but equally it is a testament to the beauty of these concepts that every time you want to go through it once again simply to understand more. This was a fantastic video - probably the most beginner friendly of all the videos I saw in this area!
I loved this stuff so much when I was a young computer science student. Finite fields, coding theory, polynomials. Heck yes.
They taught abstract algebra in an undergrad CS program?
@@FsimulatorX abstract algebra is like 90% of math done in computing
@@johnnypiquel2295 interesting. I've heard that it's helpful if you want to program new languages and such but I'm not sure to what extent it might be applicable in ones software related job (although I suppose that depends on the type of role).
Either way I'm planning on taking it in the future just out of curiosity. Extra bonus if I can apply it to one of my projects :D
@@FsimulatorX it’s probably not directly applicable to most software roles. Indirectly, though, there’s the fact that there’s a commutative ring over bit vectors of length n, with bit-wise XOR as the additive operation and bit-wise AND as the multiplicative operation, although that probably doesn’t affect most programmers. There is an algebraic structure that even the most junior of programmers has an intuitive understanding of, though, and that’s monoids. Integer addition is a trivial example, but the more instructive case is strings, with concatenation as the binary operation. It’s plainly associative, and the identity element is just the empty string. Any programming language with generics and interfaces (or traits, or protocols, whatever they end up calling it) is capable of representing monoids at the type level, although it’s really only in richly, statically-typed functional languages like Haskell where you see algebraic structures like monoids being actively modeled as interfaces that any function can be defined in terms of.
This is the most perfect introduction to Galois Theory that I have seen over several decades. It gives us not just the bare bones of the theory, but also their subtlety, their power, and beauty, and every idea copiously illustrated by clear diagrams and algebraic formulae. However, there must be something wrong with my pc, or my old ears, as I can hardly hear the voice over the music. I wish I could turn the music down, down to zero, and then I would enjoy the video for its full worth!
No it's not your pc or your old ears. The music overpowers the narration at times.
OUTSTANDING JOB! This gentleman created a masterpiece! He actually explained the Fundamental Theorem of Galois Theory in 25 minutes. Professors spend multiple semesters trying to explain what Aleph O clearly elucidated in less than half an hour.
Being able to explain things clearly is a gift few people possess.
Really missed a lot these videos. Thanks for coming back!
Finally a video after 9 months! Feels great man
Less than 5 minutes in watching and I have made more Google searches than required by an assignment. I love Maths and Engineering.
Keep up the good work.
The thing that popped out at me when I finally understood the usefulness of this:
It was not at all clear that Q(zeta) should necessarily contain Q(sqrt(5)).
I can see why such a resemblance might exist given that zeta is the 5th root of unity, but this was not at all obvious.
But, we'd already seen the subgroup of Q(zeta), it pops out very clearly in the table! and this alone is enough to prove the field contains some subfield.
I'm not convinced of this tbh. I don't see the connection between 1^(1/5) and sqrt(5). I don't think a field extension Q[zeta] contains sqrt(5), I don't see how you get there with algebraic operations
@@bobtheblob728 Indeed this is tricky to see and I’ve spent about the past hour trying to convince myself of the fact! For brevity, let’s let z denote the fifth root of unity mentioned in the video, namely the polar point (1,2pi/5). It turns out that the rectangular form of this point involves sqrt(5). In fact, you can check that 1+2(z+z^4)=sqrt(5). So, any a+bsqrt(5) in Q(sqrt(5)) can be written as a+b(1+2(z+z^4)) in Q(z), which is why Q(sqrt(5)) is indeed embedded in Q(z)! Hope this helps!
If you’re curious, there is slightly more to be said! It is also true that sqrt(5)=-2(z^2+z^3)-1, which gives us another way to see the embedding. Moreover, the reason that Q(sqrt(5)) corresponds to the subgroup that it does is because the subgroup contains the permutation that swaps z^2 for z^3 and z for z^4, which just changes the order of addition in the two equivalent expressions for sqrt(5) given above. But, as addition commutes, these swaps preserve the expression being equivalent to sqrt(5). So, we see that that subgroup contains exactly those permutations which fix sqrt(5), which again explains the correspondence to Q(sqrt(5))!
@@ronaldhoagland9597 ohh that makes sense I didn't check to see what sin/cos (2*pi/5) were. super interesting that sqrt(5) shows up there
@@bobtheblob728 You can easily check it in Wolfram alpha.
If you do the fifth root of unity (e^(2πi/5)) the number you get is -1/4 + sqrt(5)/4 + i*(sqrt(2*(5 + sqrt(5))/4)
If you want to add a root to a field you have to do a field extension. First you extend Q to Q(sqrt(5)) and then you extend again to get Q(sqrt(5), sqrt(2*(5 + sqrt(5))))
Luckily, the fifth roots of unity arrive if you extend Q by radicals (they are the solution of a quartic polynomial).
This is actually a really interesting comment. It's not at all obvious that Q(zeta5) is the only subfield in there- the only way (that I know of) to show that is via Galois theory!
As someone who has no experience in the more abstract side of math, this video was surprisingly clear!
oh wow, we have very similar profile picture
neato
this channel is extremely underrated, some of the best math content on youtube. no other vid has ever gave me as good of an intuition for this topic, and i've seen a lot of them
One day this channel will get the recognition it deserves, keep at it!
This is amazing! I have had a difficult time trying to find a good explanation of Galois theory, and this finally made it click. Thank you so much!
Thanks Jason! Glad you found it helpful :)
Beautifully crafted content.
How can one not love math or at least sense its underlying beauty? This video of yours really showcases how wonderful math can be. Thank you!
Never learned galois theory in school, just some basic group theory and field theory. I always imagined it was very daunting, but this 25 minute video was very easy to follow and gives me a sense of why people even care about this field. Thank you
The connection to unsolvability of the general quintic:
Suppose x is a root of an irreducible quintic polynomial, and x is expressible by radicals. Then we can "build up" to a field containing x with a sequence of fields like Q < Q(a) < Q(a,b) < Q(a,b,c), where each step we adjoin an nth root of some element of the previous field.* Each step's Galois group will be a cyclic group. Using the Galois correspondence, this means the Galois group of the last field will have the property that it has a sequence of (normal) subgroups where the quotient at each step is a cyclic group. This is what we call a "solvable group".
However, the Galois group of a general quintic polynomial is the symmetric group S5, which does not have this property. When you try to form a sequence of subgroups, you run into the alternating group A5 which doesn't have any nontrivial normal subgroups. Hence we have a contradiction, so the original assumption that x was expressible by radicals is false.
*e.g., say x = sqrt(2)+sqrt(3+sqrt(5)), then we'd do Q < Q(sqrt(2)) < Q(sqrt(2),sqrt(5)) < Q(sqrt(2),sqrt(5),sqrt(3+sqrt(5))) so x is contained in the last field in the sequence. You can do this for any radical expression for x.
A lot of this is not quite familiar enough for me to join the dots in your concise explanation, but I'm glad you added this, since it was presented as one of the motivating ideas of the video.
Yea do you think anyone understands what you said there..do you even understand it honestly??
@@leif1075 if you have taken a course in group theory it is much easier to understand. I have no doubt the person who wrote this comment understands it
Very lovely addition by this person to the already amazing video. I had been meaning to study field theory, group theory etc and this is one of the first few videos that came up as a result. Even if we don’t fully understand the whole thing I think it’s fine. This universe has a whole plethora of things that are so beautiful but yet we understand nothing about them. But yet, even making the attempt and going through the motions is so rewarding just in and of itself!
May be after repeated attempts - we will be able to develop the intuition for these higher order abstract ideas.
I don’t think anyone can understand it fully on the first pass!
Is it fairly easy to explain why "Each step's Galois group will be a cyclic group"? This is where things get fuzzy for me.
If there had been RUclips in my teens, I would have studied math at college. Thanks for posting! I find this enormously interesting and satisfying to watch in my middle age.
One of the clearest and most elegant presentation of Galois theory I have seen!
I would like to take a moment to thank you, and all the incredible math explainers on youtube for making such clear, and well made content. For some reason, I decided I wanted to learn Galois Theory, and I have been doing nothing but trying to understand it for an entire week now. I have come further than any teacher could have brought me thanks to content like yours. Even though I am only a freshman, I am seriously considering studying this theory even further, in order to see all of it's power. I was not only intrdocued to group theory, but to all of algebra thanks to content like yours, mate, you're amazing, and it is thanks to you that algebra is beautiful.
Thank you mate
It's always a treat to see a video from this channel. No other channel gets me as invested in modern mathematics like yours. I'm in my undergrad for physics, but I'll probably take my school's graduate algebra sequence starting next fall because of this. Keep it up.
This is beautiful! I love how you started with the concept of the root-two conjugates. Such an elegant introduction to the deeper math. Fantastic presentation as always!
Wonderful, clear videos! Great! So appreciated! At 19:47 there appears the incorrect equation "zeta + zeta^2 - zeta^3 -zeta^4 = SqRt[5]", which should read "zeta + zeta^4 - zeta^2 -zeta^3 = SqRt[5]". What follows in the video becomes correct after this revision. Small details.
Thanks! This confused me quite a bit. Also, it would help to explain in what sense and why Q(\sqrt 5) is between Q and Q(\zeta).
This will be a legendary video
this video is genuinely amazing, please more content like this! This was just the right video length and buildup for the topic at hand, i could follow every step and im about to look up some more Galois Theory, bc im genuinely intrigued now!
Never thought I would ever be able to comprehend anything in Galois Theory. Thank you for letting me prove myself wrong!
This is my 3rd watch through of this video. I still love it. I still only 80% grasp it.
Life goal: understand this video.
I had nearly given up on learning Galois Theory, but your videos gave me the motivation to continue!
AGHHHHHHHH I WAS WAITING FOR YOU TO UPLOAD SOMETHING 😭😭😭😭
Great video. I might have to watch it a couple more times to grasp everything in it, but still, it's the best explanation of Galois Theory that I've found anywhere.
Very nice video, thank you!
The only thing that confuses me a little, is when you say @15:20 that your mind can not make sense of what you're seeing. It is confusing me, because to me it is not just very logical and feels perfectly normal, but I've paused the video and could guess the rest of the permutations (after the first 4). Now I did a lot of group theory before, symmetry of groups, chemistry, knots, Rubik's cube, permutation matrices ... but it really is very natural to me, especially visually.
Anyway, nice video and keep it going ;)
Yep, I think watching Mathologer helped with that. For example the 4th one along where z3 points to z5 and back. The z1 points to z7 so this is the 7th power permutation. z3 to 7th power is z21 and 21 is 5 mod 16 so it goes to 5. z5 to 7th power is z35 which is 3 mod 16 so that goes to 3 again. All the others work like that.
Wow I recently watched Borcherds’ Galois theory series and this elucidated so much in that. Incredible video!
Minor error at 11:43. The elements of the field also have the term square 5-th of 1 elevated to 4, one term more
Actually this is not the error. The error occurs a few seconds later at 11:46. It is redundant to include this term, so it is correct at 11:43, incorrect (or at least redundant) at 11:46
Was curious if the 4th power was missed or redundant.
1 + z^1 + z^2 + z^3 + z^4 = 0
therefore z^4 = -1 -z -z^2 -z^3
so no multiple of z^4 required as a seperate term.
Later, he doesn't say that that expansion is unique, does he? He just applies sigma, correctly.
So I would not call it an "error" in either case.
But good to note that, I wondered the same.
It's a good day when this channel uploads
You got me! After reading four books of Galah theory and group theory, this was the best introduction ever thanks a lot😅
Could anyone please explain why there are only 8 roots of unity in the example with zeta = e^i2pi/16 ?
I dont exactly understand what makes the other roots redundant :)
Good question, that was also my first thought: Why only take the uneven exponents?
I'm going to watch this over and over again until I finally have an intuitive understanding of this theory. I did this in a Uni course but never got a good grasp on how we really arrive at the final result.
Trust me (as someone who doesn't live up to this advice nearly as much as he should): the way to really understand it isn't (just) to repeatedly watch things, but to find or (once you've begun to really grasp the idea through use) pose problems to play with; if you don't *use* the syntactic tools to navigate a constraint-wise consequential (i.e. well-defined) context in such a way that failure to *understand* the concept will more or less reliably ensure failure to resolve the problem - which sounds very negative, but what's important is the contrapositive supposition (which likewise seems to be the case in practice, at least up to fairly nuanced arguments about what really constitutes "understanding") that if one successfully resolves a nontrivial quantity and/or variety of examples, then one *must* possess some minimal degree of genuine understanding of the salient concepts (i.e. the ones indispensable for the problem's rigorous and persuasive resolution).
In my experience, this is the hardest part about self-studying higher mathematics: not so much access to problems, but access to *feedback* with respect to solutions the consistency/coherence/general quality of which is non-trivial to determine lol but it's certainly easier now than ever before, at least. Hope none of what I said came off as condescending or pretentious or anything like that, I just feel for anyone who also yearns to understand these things and want to help them in any way I can frankly, so...godspeed my guy 🤟🏻 haha
@@TheAgentJesus I didn't read all that but yes, I'm not only going to watch it I will play with the ideas myself as well. I'm not a media zombie
@@BachelorChowFlavour hey man I wasn't trying to insinuate any such thing, like I said sorry if it came off that way I only say as much because it's a mistake I personally have made which has affected me
@@TheAgentJesus You hit the nail on the head as to a common stumbling block for students, ie, lack of "access to feedback with respect to solutions the consistency/coherence/general quality of which is non-trivial to determine". Your comments indicate you truly understand what it takes to learn a difficult subject effectively. Kudos!
Thank you for returning! I'm so pumped! Love your stuff, dude
you really are amazing to share advanced knowledge and boil it down to something way more understandable. Keep it up!
Over at about 12:30, he skips a couple steps which confused me a bit. I'll use s for sigma and z for zeta here.
According to his map, we assumed that s(z) = z^2 (since that's where the arrow points to). Also keep in mind that z^5 = 1.
s(z^2) = s(z)^2 = (z^2)^2 = z^4
s(z^4) = s(z)^4 = (z^2)^4 - z^8 = z^5 * z^3 = 1 * z^3 = z^3
Those were his calculations.
Why does -z^8 appear?
@@3Triskellion3 Maybe it was a typo and was supposed to be '=z^8' instead of '-z^8'
Thank you so much, may I ask why does z^5 evaluate to 1?
A lot of new videos discuss Galois theory. This is by far the most profound and pedagogical discussion on Galois Theory out there.
Wow - I'd made several (admittedly rather half-hearted) attempts to figure out what Galois Theory was about over the years. Made no progress whatsoever !!! Anyway, this video actually made it all start to make sense. Truly remarkable how well you explained it all. Many thanks !!!
I'm so looking forward to re-viewing this. Great job!
This is such a cool video, I hope you keep making videos! The videos on your channel are some of the best videos on math RUclips.
This is better than other who had higher viewers..this is worth to watch..because there is no negative
Been waiting for you to post a new video! Takes me back to my first encounter with this in college. Great content as usual!!
Just amazing. You truly moved my heart with this beautiful exposition. I wish sometime to have such understanding in any field.
Amazing job.
15:25 "The pattern seems almost random"
Really? To me it seems very organized, symmetric!
P.S.: Brilliant video!
Yeah, where he sees random I just see modular exponentiation. But that's what you get from studying cryptography for as long as I have I guess. :)
Yep, I also was quickly looking at the order of each element and thinking most likely they were groups.
@@ingiford175 exactly! Pretty neat that it naturally emerges like that by studying solutions to polynomial equations.
beautiful and eloquent explanation, as always
When I watch your videos I understand close to nothing but I still love them
one of the clearest video introducing Galois group
As a physicist this intution is really helpful for me.
Wow. I'm at a loss for words. That simple question of which permutations preserve algebraic relations is such an interesting question and made everything click for me. Galois really was way ahead of his time.
Thank you for an amazing video
Ohh my god,I missed you a lot,you had a really amazing content it helped me a lot
This is by far the best video on Galois Theory I have seen on youtube. Wish I had your videos back when I was in school 😅
I really enjoy all your videos. Thanks for putting in the work
Oh my... what a well put together video on Galois's theory. My textbook makes so much more sense now
Finally after so many days
Amazing how deep these concepts can go
Amazing refresher on Galois theory after learning last year!
Thank you for sharing your knowledge and for the outstanding way you do it
God I wish I could understand more of this, hope to come back one day and see the beauty of this explanation in the same way I could finally understand the beauty of the Stokes' theorem on manifolds. Keep the excellent job, you are inspiring!
If you enjoy this type of stuff, pick up "Infinity and the Mind" book by Rudy Rucker. One of my favorite books on the math subject. I can't recommend it enough!
Wow I'm the opposite. I get Galois theory but wish to be good enough to understand Stoke's theorem!
That was a lovely video. Thank you for all your hard work and the educational content.
Thank you for your masterful clarity!
I forgot about these videos for a sec!! This upload was a surprise in a good way
This video is a true jewel indeed! Great content Sir!
Man you are finally there again, Please don't let us wait too long...because without your videos
RUclips math space looks lot less a better place!
An amazing visualization and explanation. Great Work.
Nice work!
One suggestion for improvement: when drawing roots of unity you state that they are equally spaced, but the drawings are sometimes very off. (2pi/5 sometimes looks like pi/2 and sometimes like pi/4). It could also help if you draw zeta^0=1.
I'm here from Vince's Bandcamp. I'm intrigued by this explanation of the theorem of Galois as well as the background music. You have earned my sub.
Excellent presentation and beautifully produced.
Best explanation I’ve seen
Yes! I've been looking forward to this!
I thought I knew some of this stuff, but the way you gave insight into it is brilliant. Thanks!
This video is truly amazing ! I didn't imagine someone could explain so clearly and in only 25 minutes the roots of Galois Theory.
I still don't understand everything in this video, but compared to most graduate level textbooks, this is a gift from God.
So glad you are back!
Holy smokes what a wonderful video! Thank you for taking the time and effort to produce such a masterpiece.
Wow...! I must have seen this video no less than 10 times through the years, the first while I didn't even know what a bijection was. Slowly but surely I learned everything to understand this. Thank you for being in my mathematical journey!
A very helpful and beautiful explanation for someone like new to subject . Thanks ☺️
I wish you make more such videos .
Thank you for this. Beautiful mathematics never ceases to amaze!
Omgggg I needed this so much for my exam last year! =''(
Bro where were you? Is everything all right? And why some of videos are no longer visible.
this is the most beautiful thing i've ever seen
hand down the best explanation ever
Excellent video! Can’t wait for more!
Just, too good. Really motivated to study abstract algebra now.
Awesome video on Galois theory and automorphisms of group members.
This video is absolutely brilliant!
very good video, loved watching it unfold and now i have a better understanding of what the heck galois theory is!
Thank you very much for the new video!
Beautiful work. thank you!
This video deserves way more views
That was divine and wonderful and i thank you for your intelligent work.
This is great i kept running into galois theory in my exploration of wolframs work and didnt really understand what it was until now.