 This lecture is part of an online graduate course on Galois Theory and will be about separable extensions of fields, so k contained in L. So let's just stop by explaining what a separable extension is. So a polynomial with coefficients in k is called separable if it has no multiple roots, not in k but in some algebraic closure of k. Now this doesn't depend on any sort of choice of algebraic closure or anything like that because if the polynomial is f, this is equivalent to saying that f and its derivative are co-prime in the ring k of x. This is very easy to check for example if f is equal to x minus alpha squared times something or other then you can see the derivative is equal to x minus alpha times something and therefore f and its derivative are not co-prime over the algebraic closure and therefore they're not co-prime in k of x. You can also show the converse very easily. So in an extension k contained in L, an element alpha in L is called separable over k if it's the root of a separable polynomial which we may as well assume is irreducible. This is a separable polynomial in k of x. And finally the extension k of L is called separable if all alpha in L are separable over k. So separability is just a way of making sure that the polynomials you're interested in don't have multiple roots. So in the previous lecture we discussed normal extensions and in the next lecture we're going to discuss Galois extensions and Galois extensions turn out to be the same as extensions that are both separable and normal. So this is the motivation for separable extensions, we're going to need them next lecture. We can sort of contrast separable extensions and normal extensions. They're sort of not quite complementary conditions but so k over L, let's say when it's separable or normal. Suppose f is the irreducible polynomial of some alpha in L. Then L being normal implies all roots of f are in L. And L separable implies all roots of f are distinct. So if L is normal plus separable this implies f has n distinct roots in L where n is equal to the degree of f. And so this will be very useful when we're doing Galois theory to know that polynomials have n distinct roots. So let's see some examples of extensions that are separable and some examples that aren't. It actually takes a bit of work to produce extensions that are not separable because all the obvious ones even think of in fact are separable. So first of all we notice that if k is an extension and the characteristic of k is equal to zero then L is separable over k. And that follows because if f is irreducible and the characteristic is equal to naught then the degree of the derivative of f is equal to the degree of f minus one. Assuming f is not constant of course. And in particular this means that f prime and f are co-prime. Because if f is irreducible and then it can't have a common factor with something that's of degree one less than it because it's only factors have degree or constants or the whole of f. So this means every element is separable because irreducible polynomials never have distinct roots. Well what goes wrong in characteristic p? This fails in characteristic p greater than naught because the point is the degree of f prime can be less than the degree of f minus one. You know if f is an x to the n and so on then f prime is n a n x to the n minus one which will normally have degree one less than n except something funny might happen. n might be zero if the characteristic is greater than naught that's because p might divide n. In particular we see that we might have f f prime is not one for f irreducible and the only way this can happen is if f prime is actually equal to zero and this can indeed happen for non constant polynomials f if we take f is equal to a n x to the n p plus a n minus one x to the n minus one p and so on plus a naught. So if all exponents are divisible by p then f might be irreducible and yet still have multiple roots. So let's see an example where this actually happens. Here we're not going to start with the field k. We're actually going to start with the field l which we're going to take to be the set of all rational functions over some field little k where the characteristic of little k is greater than zero and now we're going to take our field big k contained in l to be equal to k of t to the p and it's not difficult to check that this is an extension of degree p. In fact, t satisfies an irreducible polynomial. t is a root of the polynomial x to the p minus t to the p and this is in the field of polynomials whose coefficients are in the field generated by t to the p. Now you can see this polynomial is irreducible over k of t to the p but over k of t it factorizes x to the p minus t to the p is equal to x minus t to the p. So you see all roots are the same and you can also see its derivative is in fact zero and so on. So this is an example of an extension that is not separable and non separable extensions didn't actually occur in the early days of Galois theory because all the extensions they looked at at first were separable. So first of all they looked at extensions of the rational numbers and as we've just seen those are all separable and the other thing they looked at was finite fields. So all extensions of finite fields are separable and this is quite easy to see because if you've got a field f p to the m contained in the finite field of order p to the n then all elements of this as we saw earlier are roots of the magic polynomial x to the p to the n minus x. And now we notice that this polynomial is separable. You can see that either by noticing that it's got p to the n roots that are the p to the n distinct elements of this field or you can just look at its derivative so f prime of x is just equal to minus one so the highest common factor of f prime and f prime is obviously f prime and f is obviously the ideal one so these are co prime. So all extensions you come across tend to be separable unless you're looking quite hard for inseparable extensions. There's a sort of opposite of separable extensions. These are called purely inseparable extensions. So what's a purely inseparable extension? Well a purely inseparable extension is one such that any alpha in L is a root of x to the p to the n equals a for some a in k and n greater than or equal to one. Here p is of course the characteristic of the fields you're working over. And over an algebraic closure this factors as x minus the p to the nth root of a to the power of p to the n. So these polynomials only have one distinct root over an algebraic closure so there's far as possible you can possibly get from being separable. So what do you do if you've got an arbitrary algebraic extension? Well you can sort of break it up into a separable extension. Let's call this separable and a purely inseparable extension. So this is going to be a separable and this is purely inseparable. And it's quite easy to construct this. The separable extension just consists of all elements of L that are separable over k. And you can check that these form a field and that's separable and you can also check that L is then a purely inseparable extension over k. So all algebraic extensions can be divided into separable extensions and inseparable extensions. Galois theory mostly only looks at separable extensions. Point is separable extensions you can deal with by using Galois groups whereas purely inseparable extensions the Galois group always turns out to be trivial. Actually there is a non-trivial notion of Galois group for non-separable extensions. However there's a complication because the group is no longer a finite group but something called an algebraic group scheme. So if you've got a normal extension that need not be separable it actually has a sort of Galois group scheme and you can push through a sort of analogue of Galois theory for it but we won't worry about that. In this course we'll only be looking at separable extensions and only looking at discrete Galois groups. Okay so the next lecture we'll be putting together the normal extensions we talked about from the separable extensions we talked about in order to define Galois extensions which is what Galois theory is mostly about.