My favorite thing, by far, about this channel is that it started with literally "what even is a number guys" and every video has built and built and built, and now we're here talking about polynomial irreducibilitly. It feels so incredibly earned, and gives an amazing sense of perspective of exactly how ALL of mathematics builds out from extremely fundamental axioms
@mapron1 I would start with the first video he made. Truly an amazing channel that explains math well. If his videos get too complicated I would recommend 3blue1brown as a start.
@@mapron1 3b1b videos aren't always easy as there's always some level of assumed academic knowledge. Starting from the first videos in this channel, all one really needs is curiosity and the patience to think about somewhat philosophical questions such as what even is a number.
Thank you so much for these videos! My Galois Theory professor tasked me with doing a seminar on this exact topic. Glad to find an easy digestible source that isnt embedded in textbooks or papers.
Throwback to my field theory course two years ago. Amazing how even though I hated that course with a passion, this video is able to get me excited again about the subject!
You really made field theory hang together for me a lot better. Prime power galois fields, in particular, make so much more sense to me now. I could work with them before and not get things wrong, but it was all not quite as well founded. Now it's so obvious what's going on there. And in my book, if it's not obvious to you, you haven't truly understood it yet. Thanks so much for connecting all this together on a subject I never got the chance to work my way through alone.
this video is really good at helping me remember the parts of galois theory i forgot/missed the first time. like so many things in math, it makes so much more sense now
I had the first part of this video on my to-watch list for ages before finally watching it today, without even having realised that the second part was out. Happy coincidence! This leads me to wonder where the "2-ness" of our constructible numbers and shapes comes from, in a deep sense. If we add another tool to our compass and straightedge, how does that change what we can construct? What if we were fourth-dimensional beings, playing our game of geometry with a hypercompass which traces a sphere and not a circle, and a flatplane instead of a straightedge? What do the shapes that we can construct start to look like? Is that even a meaningful set of rules to try to adopt? Wantzel taught us how far Euclid's rules can take us, but what if we add new rules to Euclid's game?
@@tejing2001 oh, so that makes all the other stuff I mentioned totally unrelated. Then it seems like no matter which rules we're playing by, we can never construct a cube root.
Once again I feel the need to thank you for such an amazing video. As stated on the last video that thorem holds quite a special place for me But that is not all that makes all of your videos great, I simply love how you take your time to explain what is behind all of this math, the history, the characters, the mtivations, that makes even the most abstract problem compeling. (and once again something that I hold dear) So thanks for being such an incredible educator, one that I deeply admire and as a math teacher myself, am inspired by.
Bizarre. I'd just watched the other vid yesterday..So that is a happy coincidence. I'm assuming this is going to take me a few watches to get my head round it. Thank you.
2:30 looking at the equations, wouldn't it be simpler to cancel out the 1/4 of Gamma with the 4's it is multiplied by? Or is it written like that because Gamma, defined like that, has a particular meaning that is not apparent to me?
Wow, this is such a fascinating (and engaging) explanation of complicated subject but it puts me in mind of a question I've had since high school -- What is so important about the straightedge and compass? Did the ancient Greeks think that rulers (i.e. straightedges that you can put markings on) were too practical or unclean or something? I get that understanding the limits of the set of constraints you are working under is important but how long has it been since the straightedge was at the cutting edge of technology? We have slide rules now; heck, we even have origami (which CAN be used to trisect angles and solve cubic equations, apparently). It seems strange to me that so much attention is paid to describing the capabilities of the straightedge and compass in this day and age. Is it just because its so simple and thus easy to use as an example? Also... "the equation of a line can always be written in the form y = mx + c" -- even vertical lines?
Watch my first polygons video where I briefly discuss the origin of these types of construction (it's my latest video before this one). Just omit the y and the m for a vertical line to get 0 = x + c.
14:30 my guess is that you intend to show that 2^k dne 3^k for any nonzero k, so having the ability to create any square roots means it’s impossible to have cube roots.
Does this mean that the minimal extension to a ruler and protractor would be a device that constructs the p-th root of a number, where p is a prime? The reasoning is that the other roots could be constructed with combinations of prime roots. E.g. for the 4th root you can take the square root twice, for the 6th root you can take the square root then the cube root.
For cube roots this is true (if by minimal, you mean the minimal degree algebraic field extension we could include), but after this, for degree 5 and higher, there are many many more algebraic numbers which are not otherwise constructible. This is because in general, a degree 5 or higher polynomial cannot be solved with just the field operations and nth roots. So for example, after adding the "take 5th roots" operation, you could add the "take the solution x of x^5 + x = N" operation. This is also a degree 5 extension for "most" N, just as the 5th root of N is a degree 5 extension for most N. But importantly, the algebraic numbers given by this extension are not covered by any larger nth roots, so you really do "need" this extension, or something equivalent, for completeness. And it is the smallest degree extension needed after all square roots and cube roots, alongside 5th roots and any other degree 5 extensions.
i wonder what happens if we allow countable infinite ammount of constructions. we can construct cumulative sum of 1/1^2, 1/2^2, 1/3^2, etc. doing so we get transcendental number pi^2/6. multiplying by 6 and taking square root twice we can get sqrt(pi) and therefire we can square the circle. i wonder what is constructible like that? i think every real number can be constructed using infinite fraction expansion or something
You're right -- every real number has a decimal expansion, and every decimal expansion is just a (countably) infinite series of rationals. So every real number will therefore be constructible!
I'm confused about one point in your arguments involving field extensions - what about constructible numbers with nested square roots in their expressions like sqrt(1 + sqrt(2))? It seems like these were skipped over when you were talking about how to determine the degree of a field extension
Hello! This is counted when I talk about degree-4 extensions. Build a rational number r, then root it, so we have a degree 2 extension Q(sqrt(r)). Now build a number s *in this set*, then root it, and now I have a degree-4 extension of Q called Q(sqrt(r), sqrt(s)). (That is, assuming sqrt(s) isn't already a member of Q(sqrt(r)).) This s is any member of Q(sqrt(r)) so could be a rational or could be something of the form a+b.sqrt(r). That's how your get a nested root as you described. Hope that helps!
@@AnotherRoof I see, for some reason I got confused that the section on extensions was only referring to taking the square root of rationals. Thank you for the clarification!
So, what about squaring the circle? Are you going to tackle that one, too, or is it too far and we have to do more work before we can tackle it? My intuition tells me that it's related to the proof that pi is a transcendent number and therefore not a root of any polynomial.
I have one really big question: How can you discuss the impossibility of doubling the cube in the framework of plane geometry? It seems like a strange non sequitur that it's part of the standard discussion of the subject.
EDIT: My previous post is still valid but I made a mistake in this one. Like... If you're allowing that you can construct a cube in the first place, then you can draw a line between the opposite vertices of a cube and thereby construct the cube root of 2.
To construct cube with volume 2, what is needed is the ability to construct ³√2, which could act as the side length of the cube. That's what's impossible.
To be fair, computers can't divide by 3 either, if they're held to the standards of precision that geometry uses. You could probably trisect an angle using classic construction *roughly*, as a computer does
Thanks for creating such an informative and helpful video! I have a small question because I don’t understand one of the last conclusions. You said that if m and n gons are constructable, then an m*n gon is also constructable. Doesn’t this imply that if a p gon is constructable, then a p^2 gon must also be?
No, he said that if m and n are *coprime*, and m and n gons are constructible, then the mn gon is constructible. Notice that he uses Bezout's identity, which only holds for coprime numbers. if you use p and p, they are not coprime, so Bezout doesn't hold (the smallest linear combination you can make is the gcd, which is p, not 1). Therefore overlapping a p-gon with another p-gon doesn't give you any new lengths. I mean, just think about it. If you line up the vertex of an equilateral triangle with another equilateral triangle inscribed in a circle, do you get a 9-gon? No, you just get 2 sets of overlapping vertices, nothing interesting.
That statement had the qualification that m and n be coprime (i.e. don't share any prime factors). If your two factors are the same prime p, they share a prime factor, namely p, so a p^2-gon is not constructable.
Technically you can trisect an angle. First you have to bisect it and make a perpendicular to the bisector. Then build a right triangle using that perpendicular as a hypotenuse. We can build a 30 degrees angle so we can divide the right angle into 3 30 degrees angles. and then we draw lines from our angle to the points of intersections of the perpendicular and lines that divide the right angle. I know that I explain badly and maybe I'm wrong.
@willjohnston2959 If you can trisect 90° angle, that means you can trisect a hypotenuse of the right triangle. So we can build an isosceles triangle with the given angle. The base of that isosceles triangle is the hypotenuse of another right triangle that can be trisected. So if we can trisect the hypotenuse of the right triangle, we can trisect every angle
@@AntonDiachuk Unfortunately, it doesn't work. Trisecting the right angle in a right triangle doesn't actually trisect the hypotenuse. Even if you could trisect the hypotenuse, it wouldn't trisect the original angle. Lengths don't work that way, and school level trigonometry can be used to demonstrate this. If you're still unconvinced, get a ruler and protractor out and do it yourself. If you're *still* convinced, use this Geogebra page. The green angle is the one we want to trisect. Use the blue dots to change the angle. I built an isosceles triangle and adjoined a right triangle as you described, then trisected the right angle (which is possible). Notice that this doesn't actually trisect the hypotenuse and doesn't split the green angle into three equal parts: www.geogebra.org/calculator/cvhwydhb Trisecting the hypotenuse doesn't work either. If you're interested in why it is impossible to trisect an angle, I made a whole video about it that look hundreds of hours of my time -- I hope you enjoy it!
@AnotherRoof I see your point. The last thing I want to check when I get a chance to get to my computer if trisectors divide a base of isosceles triangles with same proportion or different. If different, then indeed it's impossible to trisect an angle
23:00 What happens for alpha equals pi, e or ln(2)? That is irrational but presumably does not have a minimal polynomial in Q, is Q(pi) considered of infinite degree perhaps?
These are called transcendental extensions, while the finite degree extensions are called algebraic extensions. Note that in transcendental extensions, each number is expressed as some rational function of the transcendental number being added, rather than simply a polynomial. This is because in algebraic extensions, we can find the inverse in terms of higher powers using the minimal polynomial, but this is not the case for transcendental extensions which have no minimal polynomial. In general, any transcendental extension of F by a single trancendental alpha, that is, F(alpha), will be isomorphic to the field of fractions of F[X] (the polynomial ring of F), which is the field of rational functions over F. This field is denoted F(X), using X as the formal variable. Essentially, transcendental extensions are a little bit "boring" but also "not nice" if all you can see is their algebraic properties. They have no special algebraic relations; that's what it means to be transcendental. And therefore you can't distinguish between different transcendental extensions with algrebraic structure alone, like a field.
@@stanleydodds9 Very interesting, thanks for writing the comment. I think it should have been more emphasized in the video the fact that inverses can be dealt with root rationalisation, something that cannot be done with trascendentals, as you point out.
Why can't we say that 8x^3-6x-1=0 is irreducible because of the following? "If it factored, the degrees would be 2+1 or 1+1+1, and so there would be a linear factor. But by the rational root theorem, there can't be one."
Yes, this would be simpler, same as with the cube root of 2 proof (he could have used the rational root theorem on the linear factor). But I think using Eisenstein's criterion here sets up the general cases where it is important, and where the rational root theorem is not useful, for polynomials of degree higher than 3 (which don't necessarily have a linear factor).
it's not cos(20), but cos(20°). it's already terrible for students to confuse these, but more so for a real maths person. you shoud know that the arguments differ by a factor of pi/180.
@@p0gr You are technically right. But my experience with "real maths persons" is that when things are obvious from context, you can generally omit them.
1:30 You can do far more with a ruler than you can with a straightedge. I mention that since you implicitly have a ruler here. The mere fact youre bringing up lengths, and in turn algebraic equations on numbers, means youve imported a metric into the plane and possess a ruler. Youve already escaped the realm of traditional Euclidean geometry.
@@AnotherRoof Im not sure if thats relevant. You cant have length without a metric. Whether that metric is defined by a "unit" segment or by a ruler is quite immaterial. My point was simply that by talking about lengths youve already escaped Euclidean geometry. Euclidean geometry had no lengths, only congruences to segments. We're beyond that now, so youd might as well use a ruler, is what Im saying. It makes no difference at this point.
@@willjohnston2959 Classical constructions dont contain a metric space, algebra, or numbers. I dont think you quite grasp my point. You are talking about lengths, which already escape the realm of classical constructions. I dont know how else to explain this to you. Seem to be running you in circles because you dont get it. Youre putting the modern notion of algebraic geometry before the classical. Im pointing that flaw out and youre fighting me on it.
Just non solvable....thought polygons of two side larger and another of two sides less, be combined through a division of the total sides. Example Equation/20x24and for the pie slices each can be averaged. is it linear for the solutions? EquationA times EquationB divided by 2. If it is not liner but a curve change the answers for the greater or lesser could be solved and it's base to establish a new count to build the two points, then the inverse, for the other two points, an x can be constructed that x will contain the lengths for said unsolved perhaps, I am new to this😂
absolutely love the production quality of your vids. you deserve a lot more recognition than you have currently, but for the time being, i am quite appreciative of and thankful for your content
The math content in this is great - but I want to specifically highlight a couple of really nice touches in here. Firstly, thank you for highlighting the spot where it's worth taking a break - for a long video like this, that's really useful. Secondly, highlighting the best place to move the subtitles to is a really nice touch (and also taught me you can move the subtitles by dragging them around the screen). Keep up the good work!
I had a moment while editing where I almost deleted these thinking "hmm, maybe people don't need these...?" but viewers have been positive about it so thanks for the feedback!
@@AnotherRoofpublish this bro. Law of contradiction p is non p Illogical impossible contradiction 1>, non p, non conscious intelligent being caused the p is non p contradiction effect of 2>, p, conscious intelligent being in the universe p is non p a false scientific hypothesis A conscious intelligent being must exist to cause the effect of a conscious intelligent being in the universe to avoid p is non p contradiction. Publish it bro. 🎉🎉🎉
This is called high quality video , discussing Maths , I think with no beautiful animations this video is still at the level of 3b1b or greater than it ... Thanks for the video broo , keep making more Also I made a video about a new calculus, "discrete calculus" Can you make a video on it in your style ?
Gasp! How can you say that most polygons are nonconstructable, when you can pair each nonconstructable one with a constructable one, and still have an infinite number of constructable polygons unpaired?!
My patrons and I were discussing this while we were drafting titles and how someone would point this out! You're right of course -- but we justified it by saying that the natural density of constructible polygons must be less than 1/2 :P
Because there are only countably infinite polygons, the only way to show a property holds for "most" cases using cardinality is if there are only finitely many counterexamples. Therefore, you need density or a similar measure to meaningfully define "most" in this context.
Could anyone explain why cube roots can't be built from rational combinations of 4th roots or higher? The proof uses field extensions of square roots but you can square root a square root etc
Good job giving a very basic introduction to the fundamental idea behind inconstructible numbers. I remember taking an introductory ring a field theory class and learning everything up to a basic introductory idea behind gallois theory, and you did a good job of getting the main ideas across without getting too bogged down in all the (important, but tedious) details.
I find it sad, that geometry and algebra are taught very separately at school/highschool. While there is this amazingly deep connection between the core foundations of both fields. Construction with ruler/compass for geometry and finding roots of equations for algebra.
You're such a fantastic educator, the way your enthusiasm comes through even with a carefully scripted video is always engaging! Lots of educational content can be hard to absorb for those of us with ADD, but you've turned what could be boring lectures into my favorite math youtube channel
Wait when did we rule out that cbrt(2) can be written using nested square roots like a + sqrt(b+ sqrt(c))? Is there some obvious reason why ruling out linear combinations of square roots is sufficient that I’m not seeing? I dont see how we answered the question from 3:17
@@debblez Ah, glad you saw the other reply! I don't think I made this entirely clear so I think I'll make a pinned comment about this. Hope you enjoy the rest of the video!
I love the way you expand on all of the specifics to do a complete explanation of things. Iff most of your viewers understand the use of logic, I'm pleasantly surprised. Beautiful belated callout for an amazing mathematician 🎉
I believe that you can't construct a cube root or trisect an angle because people much smarter than me have said so. But I have never been able to follow ANY of the proofs and I got an 'A' in geometry in school!
I've asked more than once before, about the existence and nature of a trinary operation. Like where an absolute value is a unitary operation, multiplication addition and such are binary operations. In my question, I was asking if there was any operation that took three values that couldn't be reduced to one made of components that used one and or two values... Though I didn't fully understand this video, I think it has helped me understand one of my questions. If a trinary operation existed, it's results could only be members of type degree 3 correct? As in don't exist in the number line as ones that could be made of degrees 1 and 2. Are there symbols, or generic transformations that are operations for working only with numbers that are constructions of degree 3 that don't exist in the set of degrees 1 and 2?? Or do there only exist formulas? Does there exist some kinematic device that operates on degree three numbers? Such as compass and ruler work on degree 2&4? Perhaps some tetrahedral monstrosity?
slight correction on the trisecting an angle. it is possible to trisect some angles like 30 60 90 triangles. it is not possible to trisect an arbitrary angle using only compass and straight edge.
Amazing video! Compass and straight edge constructibility was always an interesting topic for me but I wasn't brave enough to dive into the details until now. This is a very nice introduction to Galois theory. I have a few questions about some parts of the video. You showed around 35:10 that the third degree polynomial whose cos(20) is a root is irreducible (over Q). With a few extra steps, you conclude that you cannot possibly construct cos(20) with a compass and straight edge because the degree of the corresponding field would be a multiple of three, compared to the fields of degree of powers of two that we can make with the basic compass and straight edge operations. However, wouldn't this require to show that the degree 3 polynomial is irreducible over any quadratic extension of Q (not just Q)? I.e cos(20) is not a root of any degree two polynomial whose coefficients are in Q(sqrt(r1),sqrt(r2),sqrt(r3)...,sqrt(rn)) ? I feel like this is a central argument to the reasoning, but I might be missing something. Otherwise, it is possible that cos(20) would be the solution to a degree 2 polynomial with coefficients in Q(sqrt(r1),sqrt(r2),sqrt(r3)...,sqrt(rn)), making the extension degree 2 and preventing us from drawing a conclusion on the degree alone. Similarly, when finding which n-gons are constructible, you demonstrate that some polynomial is the minimal polynomial of the nth-root of unity (again, over Q). Wouldn't this also require showing that these polynomials are irreducible over any quadratic extension of Q? But then you wouldn't be able to use Eisenstein criterion or Gauss Lemma (which work for polynomials in Q or Z). I hope this makes sense.
You're right about showing the degree is 3 over *any* Q extended by square roots -- that's why we use the Tower Law. Maybe rewatch the section where we show exactly why the cube root of 2 isn't constructible. Hope that helps!
Another way would be to prove “What is the smallest angle that you can construct?” For example if you have 2 parallel lines divided into equal parts what would be the smallest angle if the lines go to infinity.
You can bisect an arbitrary angle. Thus you can construct infinitely small angles through repeated bisection so I don't think there is a "smallest angle that you can construct".
bisect a given angle into trysect a angle bisect the angle 3 times giving you quarters bisect the angle between 1/4 and 1/2 giving you the 1/3 point of the angle long way around but can be done with a compass and a ruler
It would be interesting to see a video that shows which additional polygons are possible if you have an angle trisector. I've seen in other sources that every regular polygon with 20 sides or less would be constructible *except* the 11-gon, but I can't claim to understand the math. I also know cube roots would be constructible with an angle trisector, hence it would be possible to do the doubling of a cube construction that is impossible with just compass and straightedge.
As I said I might revisit this topic in the future but, briefly: angle trisection allows us to solve cubics to form field extensions. So to see if you can make an p-gon, subtract one from it, and if the resulting number is only made of 2s and 3s then it's constructible. A 7-gon is now possible as 6=2*3, similar for 19-gon as 18=2*3*3, but not an 11-gon as 10=2*5 which would require us to solve a quintic. Hope that helps!
sqrt(r/s) is equal to sqrt(rs)/s (multiply both numerator and denominator by sqrt(s), which is of the form a*sqrt(rs) with a in Q so I think it's already covered by the product term
In my view, this is your best video yet! It might be because it coincides with my struggles to get into abstract algebra and the topics so nicely explored here serve as a great stepping stone. In any case, it's a wonderful birthday present to poor Wantzel.
Usually because Fermat Numbers are defined to be of the form 2^(2^n) + 1 where n≥0. But you're right that it feels like 2 should count; I think it's just excluded as a matter of convention.
Excellent proof! I wasn’t aware of the original proof; I had always seen it proven with Galois theory using the Galois group. And hear’s to Wanzel who died too young! 🍻
What an amazing Video!! A lot of love from Germany. Every of your videos is just amazing. Sadly this one came about a year too late since Back then I Had a Algebra course myself and the video would helped me a Lot in the field theory Part of the lecture. Keep Up the great Work, you are a huge inspiration for me and you Feed my motivation to keep on studying maths ❤️
Take an inverted prism. Fill that with water to a depth of one. Pour the contents into a jug. Fill the prism again to a depth of one. Add the water from the jug. The depth of water in the prism will equal the cube root of two. The ancient Greeks could have done that with ease.
