9.4. MORE ON ALGEBRAIC FIELD EXTENSIONS 219

Definition 9.4.13 Let G be a group of automorphisms of a field K. Then denote by KG thefixed field of G. Thus

KG ≡ {x ∈K : σ (x) = x for all σ ∈ G}

Lemma 9.4.14 Let G be a group of automorphisms of a field K. Then KG is a field.

Proof: It suffices to show that KG is closed with respect to the operations of the fieldK. Suppose then x,y ∈ KG. Is x+ y ∈ KG? Is xy ∈ KG? This is obviously so because thethings in G are automorphisms. Thus if θ ∈ G,θ (x+ y) = θx+ θy = x+ y. It is similarwith multiplication. ■

There is another fundamental estimate due to Artin and is certainly not obvious. Ialso found this in [26]. There is more there about some of these things than what I amincluding. Above it was shown that |G(K,F)| ≤ [K : F] . This fundamental estimate goesthe other direction when F is a fixed field.

Theorem 9.4.15 Let K be a field and let G be a finite group of automorphisms of K. Then

[K :KG]≤ |G| (9.23)

Proof: Let G = {σ1, · · · ,σn} ,σ1 = ι the identity map and suppose {u1, · · · ,um} is alinearly independent set inK with respect to the fieldKG. These σ i are the automorphismsof K. Suppose m > n. Then consider the system of equations

σ1 (u1)x1 +σ1 (u2)x2 + · · ·+σ1 (um)xm = 0σ2 (u1)x1 +σ2 (u2)x2 + · · ·+σ2 (um)xm = 0

...σn (u1)x1 +σn (u2)x2 + · · ·+σn (um)xm = 0

(9.24)

which is of the form Mx= 0 for x ∈ Km. Since M has more columns than rows, thereexists a nonzero solution x ∈Km to the above system. Let the solution x be one which hasthe least possible number of nonzero entries. Without loss of generality, some xk = 1 forsome k.

If σ r (xk) = xk for all xk and for each r, then the xk are each in KG and so the firstequation would say

u1x1 +u2x2 + · · ·+umxm = 0

with not all xi = 0 and this contradicts the linear independence of the ui. Therefore, thereexists l ̸= k and σ r such that σ r (xl) ̸= xl . For purposes of illustration, say l > k. Nowdo σ r to both sides of all the above equations. This yields, after re ordering the resultingequations a list of equations of the form

σ1 (u1)σ r (x1)+ · · ·+σ1 (uk)1+ · · ·+σ1 (ul)σ r (xl)+ · · ·+σ1 (um)σ r (xm) = 0σ2 (u1)σ r (x1)+ · · ·+σ2 (uk)1+ · · ·+σ2 (ul)σ r (xl)+ · · ·+σ2 (um)σ r (xm) = 0

...σn (u1)σ r (x1)+ · · ·+σn (uk)1+ · · ·+σn (ul)σ r (xl)+ · · ·+σn (um)σ r (xm) = 0