[MUSIC] Let me generalize this property to the case of fields which are not necessarily given as K of alpha. So now let me take an arbitrary and finite extension. Of K. And let me define [SOUND] the separable degree. [SOUND] A separable degree is a number of possible homomorphisms of L and algebraic closure key. Again, if L is generated by one element of five, then this degree is just the number of distinct roots of its minimal polynomial, if L is K of alpha then this is just to the number of distant root of it's minimal polynomial. [COUGH] And let me say that the extension, L is separable, over K if the separable degree is equal to the degree and I can also define the degree of inseparability. Of L over K, as the quotient of the degree divided by L over K separable. But this won't be much very important on this [INAUDIBLE], so let me formulate a theorem. First part, there so that the separable degree is multiplicative. If I have L and an extension of K, and M and extension of L. Then M over K separable is equal to M over L separable times L over K separable. And M is separable over K if and only if M is separable. Over L and L is separable over K. So this is part one, part two says that the following things are equivalent. One, L is separable over K. Two, any element of L is separable over K. Three L is generated over K by a finite number of separable elements. And four, L is generated over K by a finite number of elements and each alpha is separable over the preceding stuff, K alpha one and so on, alpha I minus one, remark the same holes. When we replace. Separability by pure inseparability. Okay, so let me give a proof of this theorem. Well, part one. We know that any phi from L to K bar extends to phi zelda from M to K bar, this is the extension theorem. And in fact, there are exactly. M over L separable ways to do this. Since K bar is also L bar, since given phi one considers K bar as L bar. An algebraic closure of K is also an algebraic closure of L once an embedding of L and to K bar is given. [INAUDIBLE] Now then the number of ways to extend is computed just by definition, by definition it's the separable degree of M over L. Now this implies the first part, thus we have M over K separable equal to a product of L over K separable M over L separable now the equivalence over separability of m to the separability of m over L and of L over K, so equivalence of separability. This is just to the inequality, in fact. Just the effect that the separable degree of an extension of our K does not exceed the true degree, okay? So if the separable degree and the degree our both multiplicative, and if one of them does not exceed another than we can conclude everything. And this fact, now this is proved as a last fact is proved by induction using the fact of that this is true for E generated by one element. We'll part two, one implies two. So one is L is separable over K and two is any element of L is separable over K. This is the consequence of the first part, the first part implies that any sub extension of a separable extension is separable. K of alpha over separable extension L, is itself separable. Two implies three. So if any element is separable over K, then L is generated by separable elements, it's clear. If any element is separable then of course, the generators are also separable, three implies four. Well, if AI is separable over K then AI is also separable over an extension of K. This is clear because the minimal polynomial of AI over K of A1, and so on, AI minus one, divides the minimal polynomial of AI over K and we have seen these in the last lecture. And so if the big one is separable, that is to say has distinct roots so then the small one is also separable. If P mim of AI over K is separable, which means that it has distinct rules, then so is it's divisor, P min of AI over K, A1 and so on, AI minus one. And finally for well, this can be proved by induction as above, I shall not give more details. So now one might ask, is the notion of separability defined for extensions which are not necessarily finite? Yes, in this case. It is best to define a separable extension as a such extension that all its elements are separable. In particular, if L over K is not necessarily finite and algebraic extension of, we can define L separable. The separable closure of K and L, as a set of all X such that X is separable over K. A separable element is by definition algebraic, of course it has minimal polynomial. So the theorem implies that this L separable is a sub field, L separable is a sub extension called separable closure of K in L. Well, I think normally one presumes that L is algebraic. K and of course, L is purely inseparable, over L separable, well and finally maybe I will repeat the important remark if the characteristic of K is zero, then any extension is separable. And if characteristic of K is P, then a purely inseparable extension has degree. P to the power R, and always this degree of inseparability is P to the power R. [MUSIC].