[MUSIC] Last time, we have defined Galois extensions. So we have L over K. It is Galois if and only if it is separable and normal. This is the definition. Of course, it's the same thing as to say that L is a splitting field, of a family, Of separable irreducible polynomials over K. And we have seen this is not a triviality, this needs a proof. But still it is equivalent to, well in the case when, L is a finite extension of K. This is equivalent to saying that the number of automorphisms of L over K is equal to the degree of L over K. I forgot to make a few remarks on normal extensions. These actually don't satisfy exactly the same properties as the other types of extensions we have seen before. So, remarks. On normal extensions. So normal extension is just a splitting field. And we have seen for instance that an extension L over M over K was finite or algebraic or separable or purely inseparable. If, and only if, it was true for L over M and M over K. So, for a normal extensions, this is not the case anymore. So let's have of extensions, K in L in M. Then if M is normal over K, then of course M is normal over L. This is clear. Since, if M is a splitting field of a family of polynomials over K. One can just consider them as being polynomials over L and say that M is a splitting field of ephemeral polynomials over L. But of course, this is not true anymore that L is normal over K. L does not have to be normal over K, right? Because L can be just a temp field over a polynomial of which M is a splitting field. Of which M is a splitting field. And when they are not equal, of course, L is not normal by definition. If for instance, A F K is Q then we can embed it into impune with the fourth root of two adjoined, and this is embedded in two Q of z and i, adjoined. This is a splitting field of x to the power of 4 minus 2, of course. And this is just a stem field of x to the power of four minus two. Which is not split in field, so it is not normal. Not normal over q. The polynomial x to the power four minus two has two roots in this extension. But it also has two roots outside of this extension. [SOUND] One more trivial remark is that quadratic extensions are always normal. In fact, a quadratic extension is a stem field of a quadratic polynomial. But you know how to solve quadratic equations and you'll see that In this case, the other root of this quadratic polynomial is also in this field. So this is just by the formula for roots of a quadratic equation. Now or else or in fact it's even easier if you have a quadratic polynomial or if P is quadratic. Over K and has one root in L. Of course it also has another root in L because P shall be the product in L of x minus its root. And something else and something else is multiple just x minus the other root. Okay? So maybe it is just useless to evoke this formula, it's much simpler than that. And this makes it easy to construct a normal extension over normal extension which is not a normal extension. One often has K in L in M where L is normal over K, and M is normal over L but M is not normal over K. Well if you take for instance, quadratic extensions, then in most of the cases, their composition will not be normal over the base field. For instance, we'll just take Q of square root of two and Q of this fourth power root of two. Both of those extensions are normal. Since those are just quadratic extensions. But Q of the fourth root of two is of course not normal over Q. So these were the remarks of the normal extensions I forgot to make last time. What we have also seen last time is L is a field. If G is a group of automorphisms of L. And over something over some key, over field, something like this. Then we can consider L G, which is the fixed field. So, these are x from L such that gx is equal to x for any g in G. Okay? On the other hand, If we have a subfield, Then we can consider the group of automorphisms. So automorphisms of L over K in the case when L is normal. because otherwise the group will be too small to give information about L. But in the normal case it makes sense to consider the group of automorphisms of L over K. So what we have seen. We have seen that if L is separable over K. Then the fixed field by those automorphisms was just K. Because of the group of automorphisms Was permuting the roots over the minimal polynomial of x over K. So, if it was fix and x was on it, was meaning that x was the only root over its minimal polynomial. So this was meaning that x was purely inseparable over k. We also have seen, That if G was finite, then L was Galois over L G. And the degree of L over LG was equal to the cardinality of G. And now we are going to summarize all these in a theorem which is in fact the main subject of this lecture course And this theorem is called the Galois correspondence. But let us interrupt just before stating the theory. [SOUND]