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224 CHAPTER 9. SOME ITEMS WHICH RESEMBLE LINEAR ALGEBRA

Proof: Just note thatK is also the splitting field of p1 (x), the product of the distinct irre-ducible factors and that from Lemma 9.4.24, p1 (x) has distinct roots. Thus the conclusionfollows from Theorem 9.4.9 or 9.4.12. ■

What if L is an intermediate field between F and K? Then p1 (x) still has coefficientsin L and distinct roots in K and so it also follows that

|G(K,L)|= [K : L]

Now the following says that you can start with L, go to the group G(K,L) and then tothe fixed field of this group and end up back where you started. More precisely,

Proposition 9.4.28 IfK is a splitting field of p(x) over the field F for separable p(x) , andif L is a field between K and F, then K is also a splitting field of p(x) over L and also

L=KG(K,L)

In every case, even if p(x) is not separable, L⊆KG(K,L).

Proof: First of all, I claim that L ⊆ KG(K,L) in any case. This is because of the defi-nition. If l ∈ L, then it is in the fixed field of G(K,L) since by definition, G(K,L) fixeseverything in L.

Now suppose p(x) is separable. By the above Lemma 9.4.14 and Corollary 9.4.27,

|G(K,L)| = [K : L] =[K :KG(K,L)

][KG(K,L) : L

]=

∣∣G(K,KG(K,L))∣∣[KG(K,L) : L

]= |G(K,L)|

[KG(K,L) : L

]which shows that

[KG(K,L) : L

]= 1 and so, it follows that L=KG(K,L).

It is obvious that K is a splitting field of p(x) over L because L⊇ F so the coefficientsof p(x) are in L. ■

This has shown that in the context ofK being a splitting field of a separable polynomialover F and L being an intermediate field, L is a fixed field of a subgroup of G(K,F) ,namely G(K,L).

F ⊆ L=KG(K,L) ⊆ K

In the above context, it is clear that G(K,L)⊆G(K,F) because if it fixes everything inL then it fixes everything in the smaller field F. Then an obvious question is whether everysubgroup of G(K,F) is obtained in the form G(K,L) for some intermediate field L?

This leads to the following interesting correspondence in the case whereK is a splittingfield of a separable polynomial over a field F.

Fixed fields L β→ G(K,L)KH

α← HSubgroups of G(K,F)

Then αβL= L and βαH = H. Thus there exists a one to one correspondence between thefixed fields and the subgroups of G(K,F). The following theorem summarizes the aboveresult.

Theorem 9.4.29 Let K be a splitting field of a separable polynomial p(x) over a field F.Then there exists a one to one correspondence between the fixed fieldsKH for H a subgroupof G(K,F) and the intermediate fields as described in the above. H1 ⊆ H2 if and only ifKH1 ⊇KH2 . Also |H|= [K :KH ].

224 CHAPTER 9. SOME ITEMS WHICH RESEMBLE LINEAR ALGEBRAProof: Just note that K is also the splitting field of p; (x), the product of the distinct irre-ducible factors and that from Lemma 9.4.24, p, (x) has distinct roots. Thus the conclusionfollows from Theorem 9.4.9 or 9.4.12. HfWhat if L is an intermediate field between F and K? Then pj, (x) still has coefficientsin L and distinct roots in K and so it also follows that|G(K,L)| = [K: L]Now the following says that you can start with L, go to the group G(K, L) and then tothe fixed field of this group and end up back where you started. More precisely,Proposition 9.4.28 Jf IK is a splitting field of p(x) over the field F for separable p(x), andif L is a field between K and F, then K is also a splitting field of p(x) over L and alsoL=Ke¢«1)In every case, even if p(x) is not separable, L C Kec 1).Proof: First of all, I claim that L C Kg(K.L) in any case. This is because of the defi-nition. If / € L, then it is in the fixed field of G(K,L) since by definition, G(K,L) fixeseverything in L.Now suppose p (x) is separable. By the above Lemma 9.4.14 and Corollary 9.4.27,IG(K,L)| = [K:L]=[K:Kgxy)] [Korey : L]= |G(K,Kea1))| [Kec :L] = |G(KL)| [Keay : L]which shows that [Kgx ,) : L] = 1 and so, it follows that L = Kgx 1).It is obvious that K is a splitting field of p(x) over L because L D F so the coefficientsof p(x) are in L.This has shown that in the context of K being a splitting field of a separable polynomialover F and L being an intermediate field, L is a fixed field of a subgroup of G(K,F),namely G(K,L).F € L=Kexn.r ¢ KIn the above context, it is clear that G(K,L) C G(K, F) because if it fixes everything inL then it fixes everything in the smaller field F. Then an obvious question is whether everysubgroup of G(K, F) is obtained in the form G(K, L) for some intermediate field L?This leads to the following interesting correspondence in the case where K is a splittingfield of a separable polynomial over a field F.BFixed fields /-* G(KT) subgroups of G(IK,F)Ky < AThen ~@BL = Land BaH = H. Thus there exists a one to one correspondence between thefixed fields and the subgroups of G(K,F). The following theorem summarizes the aboveresult.Theorem 9.4.29 Let K be a splitting field of a separable polynomial p(x) over a field F.Then there exists a one to one correspondence between the fixed fields Ky for H a subgroupof G(IK,F) and the intermediate fields as described in the above. H, © Hp if and only ifKy, 2) Ky. Also |H| = [IK : Ky].