 So suppose we have an extension field E over some base field F So we say that E is an extension by radicals over F if there exists a chain of subfields that looks like F zero sits inside prop sits properly inside of F1 Which sits properly inside of F2 which sits properly inside of F3 all the way up to Fn So we have this chain of Fields for which the very bottom one F zero it has nothing to do with Captain Falcon I mean, it's it's just gonna be the base field F right there and then Fn is gonna be the extension field E in consideration so we have all these subfields for which the intermediate subfields each have the form fi is Just an extension of fi minus one where we've joined some element alpha i Okay, and alpha i has the property that there exists some positive integer Mi that when alpha i is raised to the mi power it actually belongs to the subfield Fi minus one so I wanted to think about that for a moment if we take like for example the cube root of two Which clearly belongs to the field Q would join the cube root of two which is a Field which properly extends the rational numbers notice if I take this number So Q would join the cube root of two it is an extension of Q by it's a simple extension We just joined this element alpha i but the element alpha i has the property that if you cube it You get back to which is a rational number. So this is will be mean by a radical element So alpha i is a radical, you know square root It's a root square root cube root fourth root whatever in throat of some element of fi minus one So in this situation such an extension we say that e is an extension of F by radicals And so this is a recursive process We do allow things for like the square root of one plus the square root of one plus the cube root of two This would also be such an element Because if this is if this is our element here basically what we do is we take our field Q We extend it by adjoining the cube root of two then we extend that by adjoining the square root of one plus the cube root of two and then we extend that by Taking Q adjoining the square root of one plus the square root of one plus the cube root of two something like that, right? And so this is a this has exactly the form that we're looking for we have this chain of Fields each each element or each extension is a simple extension by a radical and what do we buy radical is that if you Take some power of this element it'll give you something back in the base field, right? So the cube root of two cubed gives you a rational number If you take the square root of one plus the cube root of two squared that gives you a number in this field and Likewise, if you take this number the square root of one plus the square root of one plus the cube root of two If you square that that gives you a number back in this smaller field So this is an example of an extension by radicals and so you can get these complicated radical like expressions We say that a polynomial f of x inside of the the base Polynomial ring f for join x we say that the polynomial f of x is solvable by radicals if it's splitting field e is a radical extension of f All right, so that's what it means for a polynomial we solve a bubble radicals So the quadratic formula is evidence that well for a ring of characteristics zero honestly it works for any every field as long as you're not characteristic to but The quadratic formula let's just focus on rational numbers here the field the characteristics zero the quadratic formula tells us that every quadratic polynomial is solvable by radicals that up to a Radical extension we have to add the square root of something to our field and that then gives you a splitting field for that polynomial now There are analogs of the quadratic formula for degree three and degree four polynomials that is cubic polynomials and quartic polynomials And while we haven't talked about them in this lecture series Maybe maybe it maybe I'll post some optional video about them sometime They're kind of cool, but but they can get kind of tedious and very difficult to use Practically speaking but be aware that the existence of the cubic polynomial does tell us that All cubic polynomial this the existence of cubic formula which it involves square roots and involves cube roots So as long as you have a field whose characteristic is not two or three the cubic The cubic formula applies and all cubic polynomials will be solvable by radicals Similar things can be said for the quartic polynomial that as long as you're a field whose characteristic is not two or three In particular if you look at care fields of characteristic zero the quartic formula tells us that all degree four polynomials are solvable by radicals some Combination of like fourth roots cube roots square roots can produce the solution to any quartic or cubic or quadratic polynomials clearly linear polynomials are already they already split since they're linear And the the quadratic formula has been known since like time in memoriam, right? We've known about this for the longest time basically At least primitive notions of the quadratic formula existed even if not the modern sense But people have been all solved quadratic equations for a long long time The discovery of the cubic Paula of the cubic formula was actually a pretty big deal because it was the first moment where the mathematical community as a whole Started to take imaginary numbers seriously. I mean there was this bigoted term We call them imagining numbers and real numbers as opposed that real numbers are more important or more real than the imaginary numbers I would beg I would beg to the argument that negative one half is just as imaginary as the square root of negative one But I digress in that situation the concept of imaginary numbers was very much You know very much mired in controversy in scandal there, right? It was the discovery of the cubic the cubic formula which use it which utilizes imaginary numbers It utilizes complex third roots of unity to solve polynomial cubic polynomial equations even if all three solutions are real numbers You can use imaginary numbers to solve cubic polynomials. And so that's really what got the mathematical Community's attention with regard to imaginary numbers was that you know, even if imaginary numbers are fake They're useful Interesting and then shortly after the discovery of the cubic equation because there was hundreds of years between This you know the formal discovery of the quadratic and the cubic goes a big a big difference there But shortly after the discovery of the cubic the quartic followed very quickly behind it It was a pretty easy discovery from one to the other the breakthrough That is the of imaginary numbers with the cubic really led to the quartic But then as people try to extend the quartic to the quintic degree five polynomials, they couldn't do it Like they couldn't find again. We got stuck for hundreds of years here Trying to find a general quintic formula and it turns out they couldn't find it because it can't be done The general quintic polynomial is not solvable in radicals and let's explain why that is It all depends upon this theorem right here. And so we're gonna prove this in the case We were which we have a field of characteristics zero. So basically like F is the rational field Q For other characteristics, you have to do something a little bit differently I'm not gonna worry about that in this video right here, but suppose we have a polynomial from a polynomial ring Where f is a field of characteristics zero then that polynomial f is solvable by radicals If and only if the Galois group of the polynomial is a solvable group Fantastic result that these two notions are in fact equal to each other They're they're logically equivalent the solvability of a polynomial has to do with the quote-unquote solvability of the Galois group Which is why we call them solvable groups by the way Because they are the exactly the Galois groups that allow us to solve by radicals This is an if and only if statement. Let's prove the first direction suppose that f is solvable by radicals Therefore there exists. Well, if you if you take its splitting field e That means e is a radical extension of f and so therefore there exists a series of Fields f0 f1 f2 all the way up to fn where f in coincides with e and f0 coincides with f Where each of these subfields each of these intermediate fields is a radical you would join a single radical element So fi is just fi minus 1 and join some element alpha i where alpha i has the property that for some positive integer mi Alpha i to the mi belongs to the base field. So it's a radical of something from fi minus 1 Okay What we're then gonna do is we're gonna consider some new fields We're gonna take the field e i which is just fi Joined by all of the roots of unity contained inside of e okay For which we want to include roots of unity if they weren't already there so it's very possible that this extension does nothing but We can throw on roots of unity and be aware that throwing in roots of unity is not such a big deal Because roots of unity themselves are radical elements if you have a root of unity say zeta n Be aware that zeta n to the nth power is equal to one which belongs to the rational field Which means it belongs to every field of characters a zero. So if you ever have a field e I which is equal to fi and join zeta n be aware that this is also an extension by radicals So throwing in the roots of unity Doesn't really complicate our solvability by radicals whatsoever, but it can guarantee that these are gawa extensions So the issue is that when you have this radical you have these this radical extension These fields might not be gawa extensions even though the radical extensions now The good news is if you throw in roots of unity that'll guarantee that each of these extensions e i over fi is a gawa extension Well, why is that because if you take this field extension right here f i and you look at this element Alpha i by construction Alpha i well, we already have f in play here take take the polynomial g i right here if you take this polynomial x to the mi power minus Alpha i to the mi. This is a polynomial inside of f i minus one a join x Because alpha i to the mi power is an element of f i minus one right there And so if you take g i this polynomial and evaluate it alpha i this didn't give you zero So it's a root of that polynomial But the other roots of that polynomial are gonna look like alpha i times zeta to the k right? These are also roots of that same polynomial Where in this case you get that zeta is equal to some you know It's equal to some mi Root of unity looks like myth right there. I hope that's not a word that causes people issues there the mi root of unity So this that then makes this extension right here e i to f i that's going to make it a gawa extension And so that's why we throw in these roots of unity roots of unity are radical extensions But so by including a few extra radicals We guarantee that we have gawa extensions where we need there to be gawa extensions So let's now look what we have right here now consider Consider the the field extensions f has contained inside of e zero, which would be f a joined all of the Roots of unity that are contained inside of e Okay, then you have e one e two So at this point what we did here is we threw in all the roots of unity that we needed From f to form e zero and then from e zero to e one We throw in alpha one then to get to e two we throw an alpha two Then we throw in alpha three all the way up to alpha n right So basically we're essentially just adding one new field to our extension here Which is just roots of unity a cyclic tonic extension, which is a radical extension And so in particular all of these Extensions are now going to be gawa extensions. So we now have constructed because we had a radical extension We were able to grow that basically by adding one new field essentially That's all we did now all of these fields are different than our original fields But if you think of the elements we have to add from We have to add to f to form e We basically have only added one new field and that includes roots of unity Which if you do if you add one root of unity at a time those are radical extensions All right, and now they're all gawa extensions radical extensions can always be turned into gawa extensions And because they're gawa extensions we can then utilize the fundamental theorem of gawa theory For which when we look at this right here as these are each each gawa extensions This this this is a normal extension of fields right e one Contains this is a normal extension of of e zero e two is a normal extension of e one e three is a normal extension of e two and each of these is a proper containment, right They always I guess that should be a proper containment right there It's always getting bigger never stays the same and so when you look at the goal the gawa correspondence This is going to flip the inequalities backwards Okay, all of these inequalities are going to be backwards now And so then you have f e naught e one e two Etc Be aware that in this situation I don't necessarily guarantee that it's proper because it could be that f already contains all the roots of unity That e contained but what have you right and so that one maybe doesn't grow This one this one definitely will grow But then when you take the gawa groups this chain is going to flip around And because each of these were gawa extensions This is going to become a sub normal series right because e one is That is since e two is a normal extension of e one that means the gawa group of e over e two Is normal inside of the gawa group of e over e one So this chain of normal field extensions turns into a sub normal series of groups Okay, and likewise The next thing I want you to note here is that because these are gawa groups Uh That that's because these are gawa extensions We have the very important property that the quotient the gawa group of e over e minus one mod out by the gawa group of e over e i That'll be isomorphic to the gawa group of e i over e i minus one That was the big part of the I mean that's one of the important aspects of the fundamental theorem That these quotients will coincide with gawa groups themselves Now when you look at this when you look at this series right here This the sub normal series. I want you to notice the factors of the sub normal series are always going to be cyclic It's always going to be cyclic and why is that? Well, because these group because these fields e contain all of the roots of unity if you would join One um of the radical elements because you already have all the roots of unity That's going to guarantee that this gawa group is actually in fact cyclic Um because this this group will have order in my that's going to be order And then you have you have an element of mi order in that you're going to have this mi cycle If you think of it as a permutation group, um, so that generates the whole group and it's going to be cyclic So we've what we've done is we've constructed looking at the gawa group here We've constructed a sub normal series for which all of the factors of the sub normal series Are going to be cyclic. All right now This is important because this series if it's not a composition series it can be refined It can be extended to a composition series because a composition series is just a maximal series, okay, and as you refine the sub normal series into a composition series these These factors will also Become smaller in that process. They're going to be quotients of this thing But the quotient of a cyclic group is always a cyclic group So as you refine this sub normal series into a composition series Since all of the original sub normal factors are cyclic This tells us that the composition series is going to be having cyclic factors as well That tells us that our gawa group is in fact solvable um as As is now just discussed right here. Um, and so This this has given us that the gawa group of our polynomial Is in fact a solvable group. It's pretty impressive how we did that now. This is an if and only of statement Um, I'm not going to prove the other direction in this video right here It's a lot harder going the other way around that is supposing that the gawa group is solvable And then proving that the polynomial is solvable by radicals That's a harder direction and i'm going to admit it because honestly We don't need it for what we're trying to show because the direction we have Is sufficient for our purposes because what we should have here is that if the polynomial is solvable by radicals Then the gawa group is solvable. That means the contrapositive is also true If the gawa group is not solvable then the polynomial is not solvable by radicals So voila consider the following polynomial f of x equals 4 x to the fifth minus 10 x square plus five This is a polynomial that we have considered already in our lecture series And in our lecture series we proved with this example that the gawa group of this polynomial is s five And as we've already proved it s five is not a solvable group because its composition series would be s five which contains a five Which contains one hence your composition factors are z two and a five a five is a simple group But it is not cyclic so Um eight s five is not a solvable group Hence by the contrapositive of the previous theorem we get that this polynomial, which is degree five polynomial cannot be solved by radicals No combination of radicals. No fourth powers fifth powers third powers third roots. That is, um You know no fifth roots fourth roots third roots square roots No combination of them will ever solve this polynomial. That doesn't mean we can't solve the polynomial It's just we can't use radicals alone to do it. It's kind of like when we talked about geometry before In this lecture series that if you talk about constructible numbers, those are the numbers which we can construct using a compass and a straight edge And we can't solve every geometric problem. We can't trisect an angle. We can't double the cube. We can't square the circle You know, those are some of the problems that we can't solve with with a compass and straight edge That doesn't mean we can't trisect angles. We just need better tools than that You know, if you have a protractor, voila, you can trisect an angle, of course You just can't do it with a compass and a straight edge alone If you have a ruler a ruler and a compass, I'm fairly certain that you can you can trisect an angle with those tools You need better tools to solve those problems. The same thing also applies to this theory of equations While radicals by themselves is an insufficient tool to solve This polynomial equation. It just means one needs better tools and these tools have been developed It's just this kind of takes us beyond what we want to talk about in this abstract algebra lecture series We want it because because this example right here is a motivating example I mean gawa himself essentially developed the modern notion of a group in order to solve this problem All right, and this is why we can name gawa theory after Everest gawa because of this fundamental idea that the solving of equations comes down to this problem With groups and it's a beautiful marriage between field theory and group theory and represents Really the climax of this lecture series So even though the problem is now proven insolvable unsolvable by radicals That doesn't mean the problem's unsolvable But it does show you the power of abstract algebra in this situation that What didn't seem like it had anything to do with each other groups having to do with this Yes groups do have something to say about not just the solvability of equations, but groups have a lot to say about Basically everything group theory can find its way into basically every discipline not just mathematics but science as well And so I hope we can take this example as the power of groups that groups are awesome And they can do everything or at least they can show us that we can't they're you know At least I like how groups in this case. We've seen examples where groups show us we can't do things But no one has to end with that pessimistic notes. Yeah It's the groups are here being used to show the limitation of the theory of radicals But groups are very very powerful tool in mathematics And so I hope you've enjoyed learning about groups in this lecture series because they're very very powerful tools And in my opinion one of the coolest things about mathematics