9.4. MORE ON ALGEBRAIC FIELD EXTENSIONS 225

Proof: The one to one correspondence is established above in Proposition 9.4.16 be-cause G(K,KH) = H whenever H is a subgroup of G(K,F). Thus each subgroup H deter-mines an intermediate field KH . Going the other direction, if L is an intermediate field, itcomes from a sub-group because G

(K,KG(K,L)

)= G(K,L) so L=KG(K,L) as mentioned

earlier. The claim about the fixed fields is obvious because if the group is larger, then thefixed field must get harder because it is more difficult to fix everything using more auto-morphisms than with fewer automorphisms. Consider the estimate. From Theorem 9.4.15,|H| ≥ [K :KH ]. But also, H = G(K,KH) from Proposition 9.4.16 G(K,KH) = H andfrom Theorem 9.4.12, and what was just shown, |H|= |G(K,KH)| ≤ [K :KH ]≤ |H| .■

Note that from the above discussion, when K is a splitting field of p(x) ∈ F [x] , thisimplies that if L is an intermediate field, then it is also a fixed field of a subgroup ofG(K,F). In fact, from the above, L=KG(K,L). If H is a subgroup, then it is also theGalois group H = G(K,KH) . By Proposition 9.4.19, each of these intermediate fields Lis also a normal extension of F. Here is a summary of the principal items obtained up tillnow.

Summary 9.4.30 When K is the splitting field of a separable polynomial with coefficientsin F, the following are obtained.

1. There is a one to one correspondence between the fixed fields KH and the subgroupsH of G(K,F). This is given by θ (H) ≡ KH . θ

−1 (L) = G(K,L). that is H =G(K,KH) whenever H is a subgroup of G(K,F).

2. All the intermediate fields are normal field extensions of F and are fixed fields

L=KG(K,L)

3. For H a subgroup of G(K,F), |H|= [K :KH ] , H = G(K,KH) .

Are the Galois groups G(L,F) for L an intermediate field between F and K for K thesplitting field of a separable polynomial normal subgroups of G(K,F)? It might seem likea normal expectation to have. One would hope this is the case.

9.4.5 Intermediate Fields and Normal SubgroupsWhen K is a splitting field of a separable polynomial having coefficients in F, the interme-diate fields are each normal extensions from the above Proposition 9.4.19 which says thatsplitting fields are normal extensions. If L is one of these intermediate fields, what aboutG(L,F)? is this a normal subgroup of G(K,F)? More generally, consider the followingdiagram which has now been established in the case thatK is a splitting field of a separablepolynomial in F [x].

F≡ L0 ⊆ L1 ⊆ L2 · · · ⊆ Lk−1 ⊆ Lk ≡KG(F,F) = {ι} ⊆ G(L1,F) ⊆ G(L2,F) · · · ⊆ G(Lk−1,F) ⊆ G(K,F)

(9.25)

The intermediate fields Li are each normal extensions of F each element of Li being al-gebraic. As implied in the diagram, there is a one to one correspondence between theintermediate fields and the Galois groups displayed. Is G

(L j−1,F

)a normal subgroup of

G(L j,F)?