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.
@Aleph0
2 жыл бұрын
Thanks for the kind words, Jim! Appreciate you watching :)
@Archer-bc6cv
2 жыл бұрын
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.
@NonTwinBrothers
2 жыл бұрын
Nah give it some time, these vids get a sizeable number of views
@900bpm
2 жыл бұрын
@@NonTwinBrothers yeah
@harriehausenman8623
2 жыл бұрын
@@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 :-)
@elidrissii
2 жыл бұрын
@@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.
@harriehausenman8623
2 жыл бұрын
@@elidrissii 🤣
@soumyasarkar4459
2 жыл бұрын
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!
@paulmeixner7445
2 жыл бұрын
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.
@singtatsucgc3247
2 жыл бұрын
If there had been KZitem 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.
@jogloran
2 жыл бұрын
Huge fan of your explanation style and visuals! Can't wait to watch this.
@p_square
2 жыл бұрын
Finally a video after 9 months! Feels great man
@Nick-wo3vi
2 жыл бұрын
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.
@ChonGeeSan
2 жыл бұрын
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 ;)
@lyrimetacurl0
2 жыл бұрын
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.
@ivanklimov7078
2 жыл бұрын
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
@AT-zr9tv
Жыл бұрын
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!
@gregorywojnar9633
2 жыл бұрын
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.
@beatn2473
Жыл бұрын
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).
@nk9083
2 жыл бұрын
One of the clearest and most elegant presentation of Galois theory I have seen!
@keryannmassin5596
2 жыл бұрын
I had nearly given up on learning Galois Theory, but your videos gave me the motivation to continue!
@FersotJ
2 жыл бұрын
Wow I recently watched Borcherds’ Galois theory series and this elucidated so much in that. Incredible video!
@MrOvipare
2 жыл бұрын
15:25 "The pattern seems almost random" Really? To me it seems very organized, symmetric! P.S.: Brilliant video!
@happmacdonald
2 жыл бұрын
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. :)
@ingiford175
2 жыл бұрын
Yep, I also was quickly looking at the order of each element and thinking most likely they were groups.
@MrOvipare
2 жыл бұрын
@@ingiford175 exactly! Pretty neat that it naturally emerges like that by studying solutions to polynomial equations.
@BachelorChowFlavour
2 жыл бұрын
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.
@SliversRebuilt
2 жыл бұрын
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
@BachelorChowFlavour
2 жыл бұрын
@@SliversRebuilt 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
@SliversRebuilt
2 жыл бұрын
@@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
@GlenMacDonald
10 ай бұрын
@@SliversRebuilt 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!
@TheOneMaddin
2 жыл бұрын
Amazing! Your best video so far! Just one note. I believe that Q has no non-trivial automorphisms. So every field extension of Q automatically fixes Q. That is, you could have saved on that technicality in the video I believe.
@dstahlke
2 жыл бұрын
I had the same thought. Sigma(3/5) = sigma((1+1+1)/(1+1+1+1+1)) = (1+1+1)/(1+1+1+1+1) = 3/5. But also it's hard to believe someone who was freshly studying this stuff to make a video would put that in there as a mistake. So I'm scratching my head...
@tappetmanifolds7024
Жыл бұрын
@@dstahlkeMonstrous moonshine group? Outer automorphisms for S6 have unique properties.
@akashsingh8502
2 жыл бұрын
As a physicist this intution is really helpful for me.
@gtjacobs
3 ай бұрын
This is an excellent video, with wonderful graphics that really make the ideas behind Galois theory come to life. There's only one thing that threw me. At around 7:42, you state that Aut F is "a little too big", so we restrict our attention to Aut F/Q. I spent a good hour trying to figure out what could possibly be in Aut F but not in Aut F/Q. At the end of it, I proved to myself that any field automorphism of any field containing Q must fix Q, so it ended up being an hour well spent. It doesn't detract from the main point you're making there, which is essentially, "this is the notation we use", and when we have Q
@MrThemastermind88
2 жыл бұрын
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!
@mallee5000
2 жыл бұрын
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!
@henk7747
2 жыл бұрын
Wow I'm the opposite. I get Galois theory but wish to be good enough to understand Stoke's theorem!
@aryamangoel764
2 жыл бұрын
AGHHHHHHHH I WAS WAITING FOR YOU TO UPLOAD SOMETHING 😭😭😭😭
@guillem2601
2 жыл бұрын
This will be a legendary video
@sharonnuri
2 жыл бұрын
Oh my... what a well put together video on Galois's theory. My textbook makes so much more sense now
@Marguerite-Rouge
2 жыл бұрын
This video is truly amazing ! I didn't imagine someone could explain so clearly and in only 25 minutes the roots of Galois Theory.
@whatelseison8970
2 жыл бұрын
I really like the music in this video. It's gentle and pensive but continuously progresses as if to say, "Consider the following: ..." Which is clearly just perfect; much like 3B1B's "Pause and ponder" music. I'm still quite a long way from grasping most of the deeper intuitions here. Still, I often find as I learn about a topic in math that my brain actually soaked up little bits and pieces of things like this as if by osmosis and then later on (sometimes much later) I'll suddenly realize I have enough pieces and enough context to understand the significance of what was being said. So thanks for making these videos and putting so much detail, care, and attention into them even though the number of people able to full understand everything in them on the first watch may be relatively small.
@TheJara123
2 жыл бұрын
Man you are finally there again, Please don't let us wait too long...because without your videos KZitem math space looks lot less a better place!
@godfreyw5412
Жыл бұрын
one of the clearest video introducing Galois group
@mirastyle
2 жыл бұрын
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 😅
@RedStinger_0
2 жыл бұрын
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.
@theboombody
Жыл бұрын
I still don't understand everything in this video, but compared to most graduate level textbooks, this is a gift from God.
@Kwauhn.
2 жыл бұрын
beautiful and eloquent explanation, as always
@Redentor92
2 жыл бұрын
Just amazing. You truly moved my heart with this beautiful exposition. I wish sometime to have such understanding in any field. Amazing job.
@thobiaslarsen8336
2 жыл бұрын
you really are amazing to share advanced knowledge and boil it down to something way more understandable. Keep it up!
@__-cx6lg
2 жыл бұрын
Nice video, quick comment - doesn't every field automorphism fix Q? It fixes natural numbers n: f(1) = 1, so f(n) = f(1 + 1 + ... + 1) = f(1) + f(1) + ... + f(1) = n*f(1) = n*1 = n So it also fixes negative integers -n : f(0) = 0, but also f(0) = f(1+(-1)) = f(1)+f(-1) = 1+f(-1), so 0 = 1+f(-1), so f(-1) = -1 So it also fixes 1/n: f(1) = 1, but also f(1) = f(n*(1/n)) = f(n) * f(1/n) = n*f(1/n), so n*f(1/n) = 1, so f(1/n) = 1/n. Therefore f(a/b) = f(a * (1/b)) = f(a)*f(1/b) = a * (1/b) = a/b. I think it's only when you need the automorphism to fix bigger subfields that you need to add the "fixing" requirement explicitly.
@lorenzobarbato4558
2 жыл бұрын
Q is always fixed by the automorphisms of a field if the characteristic of the field is 0 (so you are right in this case). Otherwise, if the field has characteristic p (prime), the field contains a copy of F_p (field of p elements) that is fixed by its automorphism group. You can show that every field contains a isomorphic copy of Q or an isomorphic copy of F_p for some prime p. I guess he didn't want to get into technicalities and forgot to mention this fact
@SurfinScientist
2 жыл бұрын
Excellent explanation! I took Galois theory in university, but my knowledge became a bit rusty after all those decades. It would have been nice if you would also have shown how this applies to the theorem that a polynomial over Q of degree 5 or more does not have a closed formula for its solutions. Maybe a good topic for a future video? For the less mathematically inclined watching this video: Galois flunked his math entrance examination to the École Polytechnique twice. His life story is quite interesting, but unfortunately far too short. It is regrettable that his genius was not recognized during his life time.
@maxwellsequation4887
2 жыл бұрын
What? You uploaded!? 9 months later... but glad you finally upload!
@pedropicapiedra4851
2 жыл бұрын
Thank you for sharing your knowledge and for the outstanding way you do it
@hellkr
Жыл бұрын
These are really heavy videos. And yet, I keep watching them
@warren64216
2 жыл бұрын
Excellent presentation and beautifully produced.
@Abhisruta
2 жыл бұрын
Hey please unlock the book recommendation video...I am obsessed with those sources...btw your videos are real 💎...thanks
@Syrian.Coffee
10 ай бұрын
Best explanation I’ve seen
@kered13
2 жыл бұрын
This was a good video but you skipped over how to find conjugacy. For example, with the sixteenth roots of unity, why are only the odd powers conjugate and not the even powers?
@japanada11
2 жыл бұрын
The easy answer: the even powers satisfy x^8=1 but the odd powers don't, so they can't be conjugate. The more in depth answer: two numbers are conjugate if they are roots of the same _irreducible_ polynomial. All sixteenth roots of unity are roots of x^16-1, but this polynomial factors as (x-1)(x+1)(x^2+1)(x^4+1)(x^8+1). Roots of different rational polynomials can be told apart from each other, so they aren't conjugate. This tells you that the sixteenth roots of unity can be divided up into five categories: 1, -1, {i,-i}, the four remaining eighth roots of unity, and the eight remaining sixteenth roots of unity. Any two roots in different categories can be told apart, but two roots in the same category are algebraically indistinguishable (conjugate).
@awaiskhan8327
Жыл бұрын
This video deserves way more views
@beauthetford7608
2 жыл бұрын
very great videos, making me seriously miss abstract algebra class. graduated before the galois theory class was offered. i'm sure i won't be able to stay away from grad school for long!
@mr.malteser5036
2 жыл бұрын
How does he reach the conclusion that sigma of zeta squared is equal to zeta to the fourth? 12:47
@jsus159
2 жыл бұрын
X2 I also got lost here
@nidalapisme
2 жыл бұрын
I got lost here too..please someone help explain 😭
12:36 he says "take this one for example." The example that is boxed is the automorphism that such that performing sigma on z results in z squared. Using the definition of automorphism from 7:12, sigma of (z times z) equals sigma(z) times sigma(z) which equals z squared times z squared, which equals z to the fourth power. Let me know if that helps.
@MiroslawHorbal
2 жыл бұрын
That was a lovely video. Thank you for all your hard work and the educational content.
@naturallyinterested7569
2 жыл бұрын
Interesting Video! Would it be possible to tune down the music when you're speaking in the future?
@alinebaruchi1936
2 жыл бұрын
Rational coeficients make what seems to be random, but it's a line of vectors in group theory and complexities. It's mostly how things work in the mind of many men who handle geopolitics, who won't act to prevent permutations of the mind that radicalize in bad ways. I found ways of fixing it
@hatimmuhammed7747
2 жыл бұрын
I really like the part when we discovered the automorphisms form a group Totally mind-blowing
@franekpiechota6514
Жыл бұрын
Great video man,I just learned what i tryed last week just beforethe exam ❤.
@jezza10181
2 жыл бұрын
Great explanation, marred by extremely irritating music
@040_faraz9
2 жыл бұрын
You are absolutely Amazing! Galois Theory is stunningly beautiful!
@78anurag
10 ай бұрын
My mind was blown at 12:50
@trisinogy
2 жыл бұрын
Awesome, but if you are going to use background music for your videos, make sure the narrator's voice is standing well above it, not just on the verge of being buried into it. My suggestion would be to remove the pointless soundtrack altogether.
@AA-le9ls
Ай бұрын
A very good suggestion!
@telnobynoyator_6183
2 жыл бұрын
He's back !!!
@gregorysech7981
2 жыл бұрын
Thank you very much for the new video!
@david8yu
2 жыл бұрын
Ive always had a fear of abstract algebra but your videos are great for explaining some of its concepts
@vmvoropaev
2 жыл бұрын
Good to see you back :)
@nihilistbookclub5370
2 жыл бұрын
Hey I love this, really I’m fucking ecstatic right now, but why did u take down ur diy math degree video?
@sunritpal9596
2 жыл бұрын
Hey that video you made where you recommended books for self study I can't find it. Did you delete it or something ? Do you mind reposting that video or making a new video of that kind. Also just wanted to say that your content is amazing. I really love it.
@wdobni
Жыл бұрын
i think that would make a great basis for a theory of quantum gravity .... where the gravitons are just the group of the permutations that let two masses in any field attract each other
@johnchristian5027
2 жыл бұрын
This video is absolutely brilliant!
@sandeshthakuri2117
7 ай бұрын
Amazing video. Simple yet profound. Doing my thesis on Galois theory. Do you have any suggestions for me? It is quite vast.
@Miguel_Noether
2 жыл бұрын
So, why there can't be a general formula to the fifth degree polynomial?
@aaronsmith6632
Жыл бұрын
Wow!!!! Can you teach us how this relates to the Quintic next video? I never took Group Theory, that is some fascinating stuff!!!
@johangamb
2 жыл бұрын
Appreciate the video, music mixed too high though - quickly got very irritating for me
@DrWillWood
2 жыл бұрын
Beautiful work. thank you!
@TheOiseau
Жыл бұрын
Sorry if this is a stupid question, but why exactly are the even powers of zeta not counted, in the example where zeta is taken to be the sixteenth root of 1? (at 14:25) EDIT : Is it because the even powers have common factors with 16 (namely 2) and thus wouldn't hit all the conjugates as they loop around the circle? In fact they would get stuck eventually (z^2 then z^4 then z^8 then z^16=1 and then you're stuck at 1 forever).
@hansoberlack7346
Жыл бұрын
The background music affects the acoustical understanding and the concentration.
@Shirokazesan
Жыл бұрын
I love the music for this video.
@dyer308
2 жыл бұрын
Very beautiful video !
@mattschoolfield4776
Жыл бұрын
You're videos are amazing sir, thank you!
@sali-math-arts2769
2 жыл бұрын
uups - just saw the answer below given by Ronald Hoagland - amazing , he writes: 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) , where z denote the fifth root of unity mentioned in the video, namely the polar point (1,2pi/5)🤓
@benjaminreynolds5733
2 жыл бұрын
Why did you delete your video on the Hodge Conjecture? Are you planning on remaking it.
@azeds
2 жыл бұрын
Just the beauty of mathematics
@elliott614
2 жыл бұрын
19:47 still trying to follow why z1z2 and z3z4 when that's not what's shown in the automorphism, where z1z4 and z2z3 And that equation would end up on the imaginary axis how could it equal root 5? If it were written z+z4-z2-z3 then we have a sum that's real and doesn't change after applying the automorphism
@user-xf6ig9ur2y
2 жыл бұрын
Nice video. Can you tell me how the example equations were developed? Just to take a guess, one could take a irreducible polynomial (with zeros in place) and multiply, add and subtract it to ones heart delight since it equals 0 and the result would again be 0.
@abebuckingham8198
2 жыл бұрын
It's actually very hard to be algebraically indistinguishable so just two different equations is often sufficient to find non-examples. You can also use the fact that if z is an n-th root of unity then z^(n-1)=1/z to find working examples.
@ClintonWatton
2 жыл бұрын
Love your videos--great job at explaining these abstract topics. Are you using Manim for your animated text?
@Aleph0
2 жыл бұрын
Thanks Clinton! This video actually just used PowerPoint for the text and animations. There's a plugin called "IguanaTex" (freely available online) that allows you to type LaTeX in PowerPoint. The animations have to be done manually though.
@notthagall7271
2 жыл бұрын
extremely useful channel
@saraswathiserene8142
2 жыл бұрын
Brilliantly explained! Thank you !!! A small aside - and I many well be in the minority here - I found the background music rather distracting.
@mrminer071166
2 жыл бұрын
This is very close to being excellent. BUT I WANT TO KNOW, and I WANT TO SEE A NICE KNOCK-ME-DOWN VIDEO OF WHY, the good permutations are good, and the bad permutations are bad!
@hyperduality2838
Жыл бұрын
Subgroups are dual to subfields -- the Galois Correspondence! Complex roots come in pairs -- duality (automorphisms). "Always two there are" -- Yoda.
@Flying0Dismount
2 жыл бұрын
Thanks for the tire rotation patterns for my car and my 6-wheeler. And the mathematical proof that they're all the same for tire wear!😜
@BharatSharma-of9du
2 жыл бұрын
Can someone tell me where's his video of a guide to learn pure mathematics ?
@nils8950UTAUACC
2 жыл бұрын
Great video! You do leave some terms you use completely unexplained though, the most unfortunate example being the notion of "conjugacy".
@CatMountainKing
2 жыл бұрын
At 11:35 where you said the points were equally spaced on the unit circle... Is that true? They don't look equally spaced to me... Edit: oh, I see, they are all one fifth of the circle away from each other, so it is including the point (1,0), or zeta^0
@derendohoda3891
2 жыл бұрын
Right, but that's a pure rational number. It is it's own conjugate, so there's nothing to swap.
@mattsmith1039
2 жыл бұрын
You posted!!!!
@ZXLegend1
2 жыл бұрын
Great vid!!
@derekduckitt5900
2 жыл бұрын
but positive root 2 cubed is 2.828 , and negative root 2 cubed is -2.828 so in some cases it does matter if root 2 is positive or negative?
@johnfist6220
5 ай бұрын
You should put more videos on your channel, because this is of exceptionally high quality but your channel is looking a bit bare.
@raffihotter4044
2 жыл бұрын
This is magical!
@rainbow-cl4rk
2 жыл бұрын
Why only the odd power of zeta when zeta is the 16th root of unity?
@harvey854
2 жыл бұрын
Because if an even power was a conjugate, then the mapping wouldn't be one-to-one. Suppose one of the conjugates was an even power of zeta, say sigma(zeta) = zeta^2. Then, for all k, both zeta^k and zeta^(8+k) are mapped to zeta^(2k). (For example, zeta^3 and zeta^11 are both mapped to zeta^6.) But, in order for a map to be an automorphism it must be bijective. In general, when zeta is an nth root of unity, the conjugates of zeta are the powers zeta^k where k and n have no common divisors. If k shares a divisor with n, then the map which takes zeta to zeta^k is not bijective and thus not an automorphism. For example, if zeta is a 6th root of unity, then the conjugates of zeta are just zeta and zeta^5. If zeta is a pth root of unity for some prime p, then the conjugates of zeta are all powers zeta^k for k less than p.
@light_rays
2 жыл бұрын
The even powers are redundant. For example zeta ^ 8 = -1 which you already have in Q and zeta ^ 4 = i which is (zeta + zeta^7) * some correction factor
@giuliocasa1304
2 жыл бұрын
@@harvey854 i was asking the same question but I've looked if it was in the comments first. Anyway where is the formal definition of "conjugate"? Can you point me to a wiki? I've found that the correct definition is maybe *primitive* roots of unity? IMO it is not a standard definition to call them "conjugates". See also the example 10.3.4 of Tom Leinster's notes: "In number theory, such an omega is called a primitive root mod p (another usage of the word ‘primitive’). For instance, you can check that 3 is a primitive root mod 7, but 2 is not, since 2 ^ 3 ≡ 1 (mod 7)." Aside from the above and more generally, he asks many times what's the purpose of this abstraction. I think that there is little if no gain speaking about fixing authomorphisms etc. when all the real concepts and the only possibilities to be clear and explain things are only based on the concrete permutations, as visually represented in this video. The abstraction is valid when it really convey a clear meaning, this modern math is an obscure, hermetic rephrase of some principles of symmetry
@lemonsavery
2 жыл бұрын
I didn't expect to be able to follow along to this (even though I'm a fan of math videos like 3b1b etc). However, this was explained quite quite gradually, and successfully guided me through. Thank you :)
@meahoola
2 жыл бұрын
Nice. Without the background music it would be fantastic.
@alexn4309
2 жыл бұрын
Absolutely great!!!
@denielalain5701
Жыл бұрын
Hi! If you keep switching the variables around in one direction, you can see that every variable is in the same place twice for each complete round. This reminds me of an electron. Can you make it so, that each variable appears more than twice at the same place for each complete round?
@ariaafrooz6749
2 жыл бұрын
AMAZING
@norepinephrined4014
2 жыл бұрын
fantastic
@drrhobert
6 ай бұрын
9:32 Now we'll use a key fact that I haven't stated this far: An automorphism sends each number to one of its conjugates. 11:38 As a set it is a set of all elements of the following form Q(\zeta)=span_Q\{\zeta^n|n=0,1,2,3\}. The last conjugate \zeta^4 can be written as as Q-linear combinations of the other conjugates due to the identity \sum_{n=0}^4\zeta^n=0. 12:06 An automorphism sends every number to one of its conjugates over Q. 19:08 Q(\sqrt{5}) is a subset of Q(\zeta) since Re(\zeta)=(\sqrt{5}-1)/4.
@HansPeterSloot
5 ай бұрын
I do not understand why at kzitem.info/news/bejne/pK2s245umnqqe2ksi=pBe3FQrs7KvQczxc&t=852 the powers become uneven: 1,3...,15. Any idea?
@drrhobert
5 ай бұрын
@@HansPeterSloot14:04 Before that let's see our final example. Here it seems Aleph 0 is implicitly considering the field F=span_Q\{\exp(2\pi i n/8)|n=0,1,2,3,4,5,6\} rather than the field Q of rational numbers. 14:19 It is the set of all elements of the following form F(\zeta)=span_F\{\zeta^n|n=0,1,3,5,7,9,11,13\}. 14:23 What are all the automorphisms of this field fixing F? 14:31 An automorphism sends every number to one of its conjugates over F.
@HansPeterSloot
5 ай бұрын
@@drrhobert Need to let it sink in I assuem It is also confusing that the last term of the formula at 14:24 is zeta^13 and not 15 so the formula has only 7 zeta terms In the example with 2/5*pi zeta^4 is in the formula and in the coordinate plot.
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