One of the few aspects of KZread that improves with time is the growing audience for content like this. It has always been a minuscule percentage of users, but with the growth of the user base there’s enough viewership to reward and motivate content creators like this. I’ll never take this for granted.

@DjordjeRomanic

3 жыл бұрын

Well said!

@antilogis6204

3 жыл бұрын

There is a broader viewership, consequently resulting in more motivation for any uploader but as you said, by force. If you think about all the crappy clickbaity content which is being uploaded over time just for the sake of some views, fame, (easy 'money'?) and it's only discouraging, so in terms of percentages I wouldn't risk it one bit. But all in all, I agree with you, good quality content can only enrichen the whole platform.

@mygirldarby

3 жыл бұрын

Anyone who was around when the internet first became available to the public knows that this kind of content was all you could get (in more rudementary, non-video form). The "average" person on the internet in the early-mid 90's was very educated and polite. It was a joy amd an honor to have experienced it.

@devanshuyelurkar1017

2 жыл бұрын

Why have you removed your learning maths video, please reupload

@hueheeuuehueuheuheuhueheeu1245

5 ай бұрын

plot twist: they are the same user base in every math channel

I agree with the sentiment that perhaps too large of a jump was taken for the relationship between the tiling and the lack of formula. I followed everything before, and that step came seemingly out of nowhere. I couldn't believe the video was already over, made me feel like something was missing.

@KemonoFren

2 жыл бұрын

There is a lot to go through to explain the specifics that a video like this can't cover. It would be fully covered in a book on Abstract Algebra.

@MA-bm9jz

2 жыл бұрын

Read about field extensions, any abstract algebra book covers this, of course maybe you lack the pre requisites

@muesk3

2 жыл бұрын

To my commentors: of course some books on the topic would cover this. That is not the point. I was saying that previous videos on this channel never made such leaps, and I could always follow them without being a proper mathematician. It was just some feedback towards Aleph 0. I still very much enjoy this video and the channel.

@leonardeuler6170

2 жыл бұрын

Yeah I agree, I could follow the general idea before the last section. But the last section was too much of a jump, took some time to see , that he is talking about tiling by subgroups and counting the numbers.

@TheZenytram

Жыл бұрын

@@KemonoFren so this video is useless, it didnt showed the Insolvability of the quintic, it explained a bit of Galois group theory and then jump to conclusion. The same is as if it was a physics channel explaining a bit the atmosphere and by the end saying "bc the sky is blue the grass is green".

Nice explanation, although I'm slightly confused since you didn't explain everything. Where do the subgroup chains come from and what do they represent? I think you should show an example with concrete polynomials. And also, although this is probably a more advanced topic, I'd like to know why exactly do non-prime numbers mean that it's not solvable, since it seems that the entire proof hinges upon this single assumption.

@mrtaurho8846

3 жыл бұрын

Solvability of groups is defined in terms of some, well, let's call them "decomposition sequences". We take the (finite) group and break it apart into smaller and smaller pieces (the pieces being appropriately chosen subgroups contained in each other) until we hit rock-bottom, i.e. the trivial group. The group being solvable is then some conditions on how exactly this decomposition into smaller pieces takes place. As it turns out (and this is in fact not so trivial) if we have a solvable group we can always take such a "decomposition sequence" and refine it (by inserting some intermediate subgroups inbetween) until every "transition number" is prime. The converse is straightforward (if all "transition numbers" are already prime there's nothing to do) and hence a group is sovable if and only if no non-prime "transition numbers" appear in the "decomposition sequence". The details are bit more delicate (as every rigorous treatment of Galois Theory is) but I hope this conveys the basic idea.

@Tsskyx

3 жыл бұрын

@@mrtaurho8846 So like, if non-primes still appear in the chain after the first decomposition, then it's not solvable?

@mrtaurho8846

3 жыл бұрын

@@Tsskyx No, only if can't get rid of them via interated refinements. So, you can start with, say, the whole group as "decomposition sequence" and then insert some suitable (not any kind of subgroup suffices here) subgroup to get a new refined sequence. If now all "transition numbers" are prime we're happy; otherwise repeat. But there are groups (notably the A_n for n>4 from the video) which just don't have any suitable subgroups to insert. Hence we get stuck at a non-prime "transition number". It's a bit hard to break down without going deep down the mathematical rabbit hole.

@Tsskyx

3 жыл бұрын

@@mrtaurho8846 I see

@yikeslikes4457

3 жыл бұрын

@@ikarienator you misunderstood the question

8:00 "In general, if a polynomial is solvable by radicals the number of tiles is a prime number." But why is this the case? This is kind of a conceptual gap in the video that left me dissatisfied. Great video otherwise!

@N9199

3 жыл бұрын

It has to do with the fact that in some sense every finite abelian group is a product of groups like Z/qZ for some q=p^k, and in some pretty specific sense this means that the number of tiles is a prime number.

@Entropize1

3 жыл бұрын

There is alot of work that goes into the insolvability of the quintic if you want a complete proof. This is essentially a culminating theorem in a first semester of Galois Theory. I recommend Milne's free online notes on Galois Theory after you've taken/learned basic group and ring theory.

@voiduniverse5563

3 жыл бұрын

Maybe this is given as a fact because he has difficulty proving it concisely or intuitively in this video. In textbook it is stated that a finite group is solvable if and only if all its composition factors are groups of prime order. If he needs to explain the fact, more details have to be involved.

@Sam_on_YouTube

3 жыл бұрын

I had the same issue.

@FranFerioli

3 жыл бұрын

This stuff has never been easy...

When this channel explains something I already know, somehow it makes me know it better

Dude. I was reading the section about galois theory literally today in Modern Abstract Algebra and I didn't get it at all lol... Anyway, thank you so much for these videos, you truly are a hidden gem of youtube

@Brien831

2 жыл бұрын

@@AmourLearning I recommend richard borcherds videos.

Bro, I hadn't checked your channel in a while and I notice there are two videos missing: -How to learn pure mathematics -Math isn't ready to solve this problem What happened to those videos:(? I really loved them.

@nahblue

2 жыл бұрын

How to learn pure mathematics is here: kzread.info/dash/bejne/mKNhw86wYtSTcrg.html It's unlisted.

@joshuajoshua46

2 жыл бұрын

@@nahblue Why did he unlist them?

I wish they had your video during my Abstract Algebra days at Uni. The "roadmap" of a mathematics course at this level is generally obscured by the cramming of fundamental concepts into our heads, not how those concepts have built the map. Thank you for filling in a fundamental part of the map.

I've seen this topic a bunch of times but never going deep enough and without this level of clarity. Massive thanks.

Great job Aleph! Been waiting for this for a while

@arindambhattacharyya5593

3 жыл бұрын

Hello there

@shohamsen8986

3 жыл бұрын

I like what u did there. Treating Aleph like his first name and Noll like his second.😂

What happened to your video guide to self studying mathematics? It was super helpful.

@shalinisharma3542

2 жыл бұрын

I know bro I am searching for it

there's nothing like the feeling of discovering a new maths channel. great content you're severely under-appreciated

The "why" you never get a prime tiling beyond 4 seems just as mysterious as to why there's no general formula for beyond quintics.

@gustavopauznermezzovilla4833

3 жыл бұрын

Actually, there is a very "algebrecity" of Sₙ for n ⩾ 5, that wraps this part out. I will copy an argument from my Galois course. Suposse that there are G₀, G₁, ..., Gₘ such that {e} = G₀ ◃G₁ ◃ ... ◃ Gₘ = Sₙ and every Gᵢ₊₁ --- Gᵢ have prime order, i.e., Gᵢ₊₁/Gᵢ is a cyclic group with prime order. Some algebra 1 theory tells us that these quotient group have the comutation property necessarily (it is abelian). Therefore, for any a,b ∈ Gᵢ₊₁, it would aGᵢ bGᵢ = bGᵢ aGᵢ and so a⁻¹b⁻¹ab ∈ Gᵢ. The element a⁻¹b⁻¹ab = [a,b] it is called the commutator, and we had seen that it transfers all the elements in Gᵢ₊₁ to Gᵢ. Okay, thats some generic bullsheet, and now we will see "why" the n ⩾ 5 its important. All those groups are subgroups of Sₙ, i.e., a set o permutations. Lets find out some dirty permutations. Chosse a,b,c,d,e ∈ {1,2,3,...,n} distinct (which can be done cause n ⩾ 5). After a lot good homework, the triplets factor as (a b c) = (d b a) ⁻¹ (a e c)⁻¹ (d b a) (a e c) = [ (d b a) , (a e c) ] Therefore, every triplet is an commutator (which we saw that trasnfers every element one step down. Using those "stepness" property on each group, every triplet in Sₙ should be in the trivial group G₀ = {e}, which is INSANE. WTF ALGEBRA DUDE?! IS THAT ALL ABOUT? SOME MIRACULOUS PROPERTIES OF AN RANDOM NUMBER? yep.

@rockinroggenrola7277

2 жыл бұрын

So, in the subgroup chain that was shown in the video, each of the subgroups creates a chain of "simple" factor groups. Factor groups are groups that you get when you "divide" successive groups in the subgroup change. Simple groups are groups that don't have any normal subgroups, a special type of subgroup. In these subgroup chains, each factor group must be simple. In the case of n = 2, S2 only has one subgroup, the subgroup of just the identity element. The factor group S2/{e} is a simple group. In the case of n = 3, S3 has a normal subgroup, namely C3, which is the cyclic group of order 3. When you take the factor group S3/C3, you get a simple group, and C3 can then be decomposed to the group with just the identity. However, at n >= 5, Sn can first be decomposed to the group An, which is the set of even permutations. However, An turns out to be simple for n >= 5, so the only subgroup which may come after it in the subgroup chain is the subgroup with just the identity. Therefore, you don't get a prime tiling from Sn to {e}. Instead, you get n!/2, which is not prime for n >= 5.

@georgecantu856

2 жыл бұрын

@@gustavopauznermezzovilla4833 so basically when n=5, Gn has multiple identities?

@artey6671

Жыл бұрын

@@gustavopauznermezzovilla4833 Dude, that's crazy. It's well-known that solvability is equivalent to the commutator chain becoming trivial, but I didn't know you can use that to easily prove that S_5 is not solvable. However, there is a stronger statement than this, namely that A_n is simple for n\ge5, where that argument doesn't work. So that cool proof is sadly not as useful as I had hoped.

I just want you to know that I was genuinely happy when I saw this in my abo feed.

I've not taken any analytic algebra courses and still able to follow along with the basics thanks to your great visuals and explanations.

I love to explore the mathematics this way! Thank you so much.

Your channel rocks! Especially liked the self-study video. Thank you!

@shalinisharma3542

2 жыл бұрын

Where that video is I can't find it

Great job!! I always wondered how that worked out. As I specialized in functional analysis and probability theory, I never got the chance to dive deeper into this stuff. Thank you!

You, sir, are another potentially big player on the mathematical KZread scene, which is clear from the quality of your content and the fact that you have almost 70K subscribers. Keep up the great content, you are doing the right thing. I like the way you present the content, reminds me of ASAP Science.

Great video! Thanks

I have no idea whats going on but, I just love ur video's.

Many thanks. I did a course on Galois Theory in the third year of my maths degree, 40 years ago. I can't recall much of it, but this video has brought bits of it back to me. This has whetted my appetite to read up on it.

👏👏👏 just made sense of a full course on Galois Theory full of formalizations of automorphisms and company and empty of intuition. Great video.

You just read my mind wow I started learning about this topic since 2 days. thanks for the great explanation and quality.

i see a lot of people in the comments here who find this explanation unsatisfying, and this is certainly a good instinct to have! the reason is however because to actually understand this result you need a courseful of context, and a 10 minute youtube video can only do so much. i don't think this video was ever supposed to be a comprehensive explanation, but just an introduction to the topic c: btw the way the music loops is killing me

Love your videos! A recent problem set of mine involved proving that A_n is simple for n>4 and this is making me excited to learn Galois theory this spring

@harriehausenman8623

3 жыл бұрын

Just don't get into any duels ;-)

Really enjoying your series of videos. Thank you.

Although there may be a detail or two that you did not explain, still you break the discussion down to the crux of the matter and thus make the discussion accessible to the uninitiated. That is inspiring! Now I really want to learn the details. Thank you 😊

I'm going to be studying field extensions and Galois Theory next week so this video is extremely helpful.

I'm looking forward for when you'll have millions of subscribers. Great presentation, unique aesthetics and great didactics ! perfect!

Another totally amazing video. Huge thanks.

great video as always, I'm looking forward to taking abstract algebra next year and this is an awesome preview thank you

Okay man I have to say that I've watched and read a fair amount of introductory content to Groups, Abstract Algebra and Field Extensions and this is by far the most comprehensive one. It is amazing how clearly you present the ideas, an statement of how organized you have the topic in your own mind, truly amazing! Thanks!

@harriehausenman8623

3 жыл бұрын

Me too! I try to tackle these ideas for a while and this kind of 'quick but still precise' recap helps a lot.

@cufflink44

Жыл бұрын

Gonzalo Christobal Well said.

Just wanna say, love your videos Aleph! This one was especially interesting and cool, really excited for when I get to group theory in Uni :)

In case anybody is interested, there's also a really nice elementary argument (no Galois theory) for the statement about a general formula due to Vladimir Arnold, which I just put up a video about here: kzread.info/dash/bejne/dId82Jt-nM6Xfbg.html Some other resources in the links there too :)

@MichaelRothwell1

2 жыл бұрын

This video by @not all wrong is amazing. Full proof, no gaps. Highly recommended! It also explains exactly how you get you get the decomposition series of a group - by taking as your normal subgroup the commutator subgroup at each stage. This obviously gives an abelian quotient, since you have set each commutator (ghg^-1h^-1) equal to the identity element.

Been waiting for your video for a long time!

Brillantissime vulgarisation ! En 10 minutes seulement les grandes ligne de la preuve et ses éléments de la théorie de Galois. Merci !

@grossetalain1234

Жыл бұрын

Tout à fait d accord. Seuls les anglo saxons arrivent à faire cela.

Great intuitive video, taking my second semester abstract alg class rn and this is spot on

Man, your video is fascinating. What an age to live in

Very nice explanation. Thank you for this excellent work!

Marvellous work!

You had a video on books to read for people who want to learn college mathematics and it had a link to an open letter for calculus, can you share it again please? Or anyone else who passes bu here and know what I am talking about.

@thirdwave1777

2 жыл бұрын

looking for the same. please post here if you find.

@everlastingideas8625

2 жыл бұрын

@@thirdwave1777 I am still looking for it but yes, I ll post it here if I find it sure. Please do the same if you can.

Very nice presentation of a difficult concept. Keep up good work.

really nice Video! I quite like the long end card (like 20sec), because i gives time to rethink everything you said, before having to interrupt the autoplay

Brilliant video!!

Very good 👍

Finally, someone can explain this quintic mystery 👏. Thank you!

Thank you so much for making this material, you're amazing 🙂👍

A very nice presentation! Thanks for referring to my book "Galois Theory for Beginners".

I love this. The explanation was simple enough to understand as a Physics student.

You do a great job.

So hyped for this one 💯💯💯

I loved Galois Theory. I wish I could continue this path, but the other mathematical subjects were too hard for me and could not pursue getting my Ph.D. I ended up taking 2 yeas self studying abstract algebra just to understand this proof. It was worth it.

@artey6671

Жыл бұрын

It's my understanding that you basically need to be a genius to get a PhD. If this proof gives you so much trouble, then maybe a PhD isn't the right thing for you.

New upload :)))) your videos are amazing

Great video!

Totally enjoyed this video!

I love your videos !

this is beautifully done.

beautifully explained !!

Great work! I love watching these videos

That's the summary of Abel's, Galiois work. Brilliant

Excellent. Thank you. Could not be explained better.

I had a surprisingly difficult time finding someone bold enough to make the simple, explicit claim that nested radicals correspond to field extensions. You'd think that would be easier to find, but I had a heck of a time finding it. Didn't see it on math stack exchange, and darn sure didn't see it in grad level textbooks because they never explain anything clearly. Didn't find it until seeing it in your video. Thank you very much.

Thanks a ton....scratching my head to understand this concept from last 4 years . Now i can make sense.

amazing work

The most impressive thing is that Galois came up with most of his ideas and wrote everything just a couple of days before he died. At the time he was only 20.

@gianfrancodue1476

3 жыл бұрын

Well, Hardy used to say no good Mathematics got discovered by someone over 50, most likely 30 too. He got a point, somehow

Please provide a full course in calculus cause you really really explain well

Your videos as awesome, very didatic. I would love to see a video about harmonic analysis, for instance about the philosophy of the Fourier transform or something about the Calderón-Zygmund operators

Two tiny nitpicks: First, at 2:31 I think it's misleading to write it as at every step a n-th roots is adjoined. This is not necessarily the case as one of your examples illustrates too. Might've been better to use n_i instead to indicate that the degree of the extension is not necessarily the same at each step. Second, I agree with the comment by @diribigal that the phrase "there is no general formula" is not exactly precise and might cause some confusion. For example, X^5-2 is a sovable polynomial of degree 5 as the Galois group is not all of S_5 but some proper solvable subgroup. However, at 8:25 you write "Gal(general degree n polynomial)=S_n" which, IMO, only adds to the confusion: the general polynomial meant here is in some variables t_1,...,t_n with no algebraic relation between them and not any polynomial over, say, Q. I know that it's hard to be this precise and still concise at the same time so I thought it might be worth adding a comment. Last but not least: I really enjoy your videos and I'm looking forward to more!

@harriehausenman8623

3 жыл бұрын

good point

@pineberryfox

2 жыл бұрын

nth root is fine actually. the square root of 2 is the sixth root of (2^3)=8 and the cube-root of something is a sixth-root of the square of that thing. in general, for finitely many adjunctions n is the lcm of the individuals. imo adding the subscripts would at least somewhat hamper readability, and because it's not necessary i'd rather them be left out

The quality of your content is exquisite! Keep going with this tempo! Btw,why did you delete the video on Hodge Conjecture ?

You used to have a great video for people who wanted to study pure maths with a syllabus suggested you tube videos and books where did it go I would love to see it again or at least have a list of the videos and books thanks

Really great video and channel!

Nicely explained

Wonderful. Thank you.

Pretty informative and understandable for distilling a whole undergrad course into 10 minutes

I am really amazed by how mathematicians and engineers are incapable of giving conceptual explanations of some fundamental ideas in their areas, but you did it!! you made me understand what solvable by radicals means, in a simple and insightful way; many thanks! the most useful part for me was the beginning, when you explained we had to add radicals step by step; I know the Insolvability of the Quintic isn't in contradiction with the Fundamental Theorem of Algebra, but then what is a root when it is not given in radicals?! this is also a thing which I think is fundamental but I can't find a good insightful nutshell explanation of;

Youre amazing

A Cambridge University math course that comprehensively explains Galois theory so elegantly in 10 Mins. Wow thank you so much. I feel cheated by my university that could never really explain anything. We were simply following procedures.

This isn't what I was looking for but thanks for that lesson.

This video is the nice introduction.

Another knockout! No fat but still clear and satisfying

Great!

Thank you very much, very informative video. Since your handle is Aleph 0, I'd love to hear your take on Cantor's countable/uncountable infinities (I'm not proponent of the idea).

Why have you removed your learning undergraduate maths video, pleas reupload

MANY THANKS.

I hope you do a video on V.I. Arnold's proof of the insolvability of the quintic. It might be easier for viewers to follow.

Awesome! (notifications ganggg)

THANKS!

I really like the style of your videos, with the handwritten illustrations. What do you use to edit and create your videos? Also, thanks for sharing your mathematical knowledge!

I hope you will make a video explaining the Yang-mills existence and mass gap. And explore some possible solutions to it. Anyway, that's a great video

Congrats. This is the most understandable and intuitive presentation of the Galois Correspondence and quintic insolubility. Contents of books are a torture to assimilate. Other lectures (lecture series) can't see the forest for the trees. Thank you

Hi.. you post great videos. I was wondering if u can suggest any book(s) on your previous video regarding Navier-Stokes Equation (how it is used in fluid dynamics)?

Thanks❤️

Wonderful

Amazing

if I remember correctly you had a video about tensors and how tensors are related to the direct summ or product or something like this. did you took it down or did i just dreamed about it because i cant find it anywhere. And you do a realy good job keep it up we all appreciate your work.

Thankyou so much

This is good enough.... if it's too long it'd be like a whole-semester course 😝 We want a quick overview and the interested reader can dig deeper.

fuck i've waited so much for some new content, you're the man!

Love your videos! Maybe this one should have been split in more videos, it feels a bit too complex and less clear compared to the previous ones.