Normal and Separable Field Extensions Explained | Abstract Algebra

[00:00:00]Okay, welcome everyone to my continuation of what is algebra. Um, today's topic are the so-called normal and separable field extensions or just extensions.

[00:00:11]And kind of what I would like to explain is why this is kind of equivalent to my slogan or roughly equivalent of course to my slogan that linear factors matter.

[00:00:22]So splitting polomials in BA and yeah the whole idea is of course that you have some certain symmetries that you would like to study in Galawa theory and well the study of field extensions has grown into an old own well field of malabatics whatever but some field extensions are better than others to do galawa theory and kind of the normal and separable extensions pick out those that that are there and yeah well the whole story basically goes as follows. So in Galawa theory so left to side field extensions themselves in Galawa theory you would like to study nice field extensions in the following sense. Um so Galawa theory is a study of symmetries on roots. I will run the Mathematica uh program in a second to show you what I mean kind of a group acts on the roots of a polomial and that's what you would like to study in in galwa theory under the umbrella of field extensions turns out that no not not all field extensions are good enough um some of them are better behaved when when you want to see those kind of group actions here um which we'll see in a second so basically those will swap in this case.

[00:01:45]Uh so the red dots will be to the roots and they will they will be permuted by by a group acting on the on the roots and yeah so the main idea for field extensions is of course that you throw in roots of polomials. Um but but you have to do that in a good way in order to really reflect those group actions.

[00:02:05]Um but be before we can get into that let me just let me just show you um what Galwa theory wants to study. So okay here's Mathematica um so this illustrates a polomial of degree n so n is two I could do is three I could do four and the red dots are uh the roots and I'm able to vary um a certain coefficient in my polinomial. So this is a degree 2 polomial and I'm allowed to vary the coefficient in front of the x variable right. So it's x squar plus a certain coefficient uh in front of x uh plus 1. So if x is this coefficient in front of x is zero that's that's when my black dot is right in the center here. Uh the two roots of x^2 + 1 are of course i and minus i.

[00:03:02]That's how you should read it.

[00:03:04]And it turns out that up to certain choices for this um for this coefficient in front of x uh this is pretty pretty pretty nice. So um if I move this coefficient then the roots as you can see move continuously. But then there are some certain crucial um crucial values as you can see almost. So the numerical precision of this program is not optimal but that doesn't matter. So basically if I would move my mouse precisely on this blue spot here I would get multiple roots.

[00:03:43]Uh similarly for this blue spot. So there are two spots where this polomial behaves slightly different depending on the coefficient uh in front of the x term.

[00:03:54]But anyway, you can also see well the point is here's a group action. So here you can see it. So the two roots swap their position. So there's a group action of in this case a symmetric group in two letters. Um we can do better. So let's increase the degree then it gets slightly nicer. So same setup.

[00:04:17]This is x to the f to the 5 plus scala * x + one. And I can vary the scala by varying this black dot. And as you can see the roots still behave very very um continuously with with when I'm moving this this dot. And the point is again if I move here then two roots would collide.

[00:04:41]Uh so you get multiple roots. And here's the action of a group. As you can see, this is clearly a permutation action of the group. So there's clearly something going on here. Things permute if I move around like this. And this is kind of what Galawa theory wants to study.

[00:04:57]Again, here if I move around like this and this doesn't depend on the degree degree 8 x to the 8 scala * x + 1. Same pattern just more roots. This is what Galawa theory would really like to study. Yeah. So back to my slides. So this polomial that you see here that I just illustrated was x to the 5 plus scala * x. So scala is a black dot plus one. And the red dots are the roots. And you clearly have an an action of of of a group on those roots even in some continuous way. And kind of that's what you want to study in in galawa the galer galwa theory. There's always a group acting and that's what you would like to see reflected in field extension and sadly that doesn't work for all field extension or maybe not sadly maybe it's expected that it doesn't work for all field extensions the zoo all field extension is just a little bit too big and this these normal and separable pick out the ones that behave nicely with respect to these actions on groups.

[00:06:05]So let's have a look at an example a very very easy polomial uh well maybe not too easy but certainly not too hard x cubed minus 2 and of course all of you know at least one root of this thing which is the uh the third cube root of two which is here but it has two other roots one is here and a blue one down here and you have this beautiful symmetry that they arrange themselves along a triangle and of course on a triangle you have a group action right anyway so um don't look too much at the expressions here the point is that's where they sit in the complex plane so in the background you have the complex plane uh one of them is a real root of course there one third root of two which is a real root and the others are not real roots so this field extension which you naturally would think is associated to this uh field here is certainly a sub field of of of are and that's kind of a problem because you have obvious group action on this picture for example you can just swap those two uh those two roots this kind of a zod 2 action on this picture by just swapping the two roots and the two roots um are always kind of scaled versions of the of what I call usually za and za is just a choice of a primitive third root of reality anyway and um Yeah. So the natural in quotation marks uh field associated to this equation is Q adjint this maybe this third root of two this real number uh this real number here. So you get a sub field of the real numbers but certainly this group action that I just explained or just showed to you here this one that that that's lost. You don't see you don't really don't see it if you just look at um this field extension here.

[00:08:03]In other words, the group action desends if you just look at field extension to the identity. You can't distinguish it from doing nothing and that's not really nice.

[00:08:15]So in some sense and yeah in in the sense of this video at least this is a bad field extension. It's an illbehaved example. I priority nothing is bad about this field extension but it doesn't really fit in this philosophy of studying group actions on roots.

[00:08:32]Um the algebraic way to see this and that's the one maybe you want to keep in mind is that the polomial itself the x cub minus 2 has one flaw that you can see without knowing anything about group actions. The flaw is it doesn't split into linear factors in my field. And the problem is um yeah I just just can't see those two roots. So I can't factor it in linear factors.

[00:08:58]And I said again this is a property that you could could observe without knowing anything about a group action. So without knowing anything about Galwa theory.

[00:09:07]So let's have a look at a well- behaved example kind of a it will be the closure of this example. We'll see that in a second.

[00:09:15]um don't look too much again at at the precise values. So here's a certain polomial which I call f and it has degree six and the point is what I do so kind of what is missing here is are the the complex solutions but kind of what's missing is is this um uh the complex number which I then just throw in. So instead of looking at just Q adjoin square root of three uh cube root of two square root of three cube root of two um I I just throw in my little little here which is the same as looking at this element. That's not quite obvious. You could think about it a little bit. It's not so hard to see that those two field extensions are the same.

[00:10:01]And um it happens or this is how this example is constructed that this guy up here is a this was really bad. Give me another one. This guy up here is the minimal polomial of uh theta plus third uh cube root of two. Very good. And this is how the roots look like. And you clearly have kind of the same type of symmetry. You have this um swapping symmetry which connects those two dots.

[00:10:32]those two dots. It's really the mirror uh along the it's really just a mirror along the x-axis and it connects those two dots and there's no information loss. You can still see it in your field extension because now all of these elements are actually inside of your field extension. So all of these roots are inside of your field extension. Uh you have another symmetry. So actually the symmetric group S3 acts here and this is this triangle symmetry. This is Z mod 3 acting here and acting here just by going around the triangle right and as you can see this gives a nice graph and everything is nicely connected and you can go from everywhere to everywhere. So this looks pretty nice and um yeah so this whole action of S3 is present in my fields extension. You can always see it. It's any element of S3 gives you a non-trivial um operation on your field. Right? So in really in stark contrast to this example where I had a non-trivial operation on my roots which was absolutely not reflected in the field at all here it's different. So this field really reflects um the um the action on the roots and the algebraic property you would observe is that f actually splits in this field. It's a minimal polomial of of this expression and you can check that its roots are of this form whatever it so this polomial does split. So even if you don't know anything about this group action on the roots, this is still a better in some sense field extension than the other one because the defining minimal relation the defining minimal polomial splits in this uh field extension and that's then exactly the definition of a field that is normal.

[00:12:20]uh later or in Galwa theory you will understand why this is called normal because it corresponds to normal subgroups of of of certain of this group acting on the roots but anyway for now it's just a normal fields extensions and it's the following so throughout this slide I'm always only looking at finite things so or only always algebraic whatever I'm not going to say that anymore so you have a field extensions L of K and it's normal if um every irreducible polomial splits with a root in L splits. So this is exactly what I just said. So this is a irreducible polomial over Q and it splits in in in this field down here. So this is L on the other slide.

[00:13:08]So it splits in L, right? And that's the only really relevant polomial. So this is this definition that polomial split and then you call it normal and galawa theory will tell you that these are the right field extensions in the s in the sense of h having having nice actions uh on the roots or let's say they're almost the right ones. There's a second condition uh which doesn't play a huge role in let's say uh if you work over Q over Q actually doesn't play any role at all but I will explain on the next slide why you should be worried in some sense about this condition and it looks a little bit strange and it is the the so-called separable so separable extensions are those extensions where any minimal polomial corresponding to your L has a formal derivative not equal to zero. I will explain that on the next slide. Um the point is in some sense if you for the first time learn uh this was really bad. Give me another one. Um for the first time learn about field extensions um and galawa theory you can kind of ignore this condition.

[00:14:23]So it's for example always satisfied if you are looking at fields extensions over over the rational numbers.

[00:14:31]There are certain we need it. There are certain field extensions that that completely is well inseparable. Um so you really need it but let's not worry about it if you walk over Q. The other one is um the crucial one or in some sense the crucial one that I showed you that really wants to reflect those uh symmetry actions of of of those symmetry groups.

[00:14:54]And um the main theorem to keep in mind about the the first one the normal extensions is that they always they always exist. So you can always make if you have a field that is too small it is too small to see those actions you can always make it a little bit bigger. So this is a normal closure of this one.

[00:15:14]You can always make it a little bit bigger such that it is normal. So it's kind of a a nice condition. It satisfies whatever you want. it exists and if you throw in enough minimality if you throw in the right minimality condition it's actually unique and unique of course means up to okay so that's normal normal is kind of well we can it's not quite clear from this definition but later you will see actually that this is equivalent to saying we can see uh well if you if you for it's equivalent if you forget separable so let's forget separable then it's equivalent to um you see all your symmetries of your um of your roots reflected in in the field extension.

[00:15:59]Okay, so this video is called normal and separable field extensions. So at least should tell you what separable really means. I I stress again if you work over over um something like the rational numbers, you don't really care. It's a non condition. It's always satisfied.

[00:16:14]But still well you will find it in textbooks and it's important if you work over well let's say fields that are not as easy as version numbers and the idea is pretty simple so it fits on one slide and the reason why we want this is we don't want minimal polomials to have multiple roots that would be kind of bad because um we want to permute the the roots but now two roots are the same.

[00:16:39]This this already sounds really bad from this viewpoint of permuting roots. And the definition is then well surprisingly straightforward. This is as follows. You take a formal derivative. If you never seen formal derivative link is in the description, it's it's the following idea. It's it's pretty simple idea. Um so if you well derivatives that sounds like calculus when we are doing algebra.

[00:17:01]So what what I'm saying here this sounds like nonsense. Well, we already learned that a polomial is not in some sense not really an an object of study of uh analysis or calculus. But you can also interpret a polomial completely formally as an element of a ring of a polomial ring. Right? So you can just say the polomial is f is an element of of a polomial ring of something like this without referring ever to any kind of polomial you see in analysis.

[00:17:34]And if you think about it a little bit, there is an operation that's taking the derivative in analysis. Um, which looks like really this is at the heart of analysis. And derivatives are surely at the heart of analysis. Derivatives and integrals. Um but if you just throw away all the analytic things and just look at the rules, you take a polomial and you calculate its derivative and you take that as a definition of your derivative, then you have just defined what is called a formal derivative and it's really what you think you would do. If you would like to take the formal derivative of x^2 + 2, for example, um so this should be here. So this is a typo. This is this is a1 not a z because you take the um derivative of x^2 + 2. This term dies and you just get take this one pull it down. So you get 2x, right? Another one x cubed + 2x for example squared. Uh you take the derivative and no matter what you do you you it's just an algebraic operation, right? you you I could tell you to take this derivative if I would give a lecture on calculus let's say I could tell you how this works without ever explaining kind of an epsilon delta definition or whatever right and that's what I do I just take exactly the same definition and it's satisfies exactly the same properties like product rule and linearity and the condition of um this funny condition that the minimal polomial has formal derivative zero is equivalent to the minimal polomial has no multiple roots.

[00:19:20]This is just a very easy thing to check.

[00:19:21]You just take the derivative of a polomial.

[00:19:26]Anyway, so let me wrap up. So um normal extensions and in the inseparable extensions as well come from the observation that field extensions are a little bit formal zoo that is a little bit too big uh to study or a little bit too wild to study galwa theory. Galwa theory is all about this action of uh of groups on the set of roots and normal and separable are kind of built such that you you have a nice action on those roots. Normal just means everything factors into linear factors. Not everything the relevant polomials factor into linear factors and separable is this condition in the end it's a condition of having non not multiple roots because multiple roots are kind of bad from the viewpoint of of an action of a group. Yeah. Well, anyway, I hope you enjoyed the video and I also hope to see you next