9.4. MORE ON ALGEBRAIC FIELD EXTENSIONS 217

Then from Corollary 9.4.6, using q(x) and q̄(x) in place of the p(x) and p̄(x) in thiscorollary, there exist k ≤ m one to one homomorphisms (monomorphisms) ζ i mappingF(r1) to K̄≡ F̄(r̄1, · · · , r̄n), one for each distinct root of q̄(x) in K̄. These are {ξ 1, ...,ξ k}where k≤m. If the roots of p̄(x) are distinct, then this is sufficient to imply that the roots ofq̄(x) are also distinct, and k = m = [F(r1) : F] . Otherwise, maybe k < m. (It is conceivablethat q̄(x) might have repeated roots in K̄.) Then by Proposition 3.4.13,

[K : F] = [K : F(r1)]

>1︷ ︸︸ ︷[F(r1) : F]

and so [K : F(r1)]< [K : F] .Therefore, by induction, two things happen:1.) Each of these one to one homomorphisms mapping F(r1) to K̄ called ξ i for i ≤

k ≤ m = [F(r1) : F] extends to an isomorphism from K to K̄.2.) For each of these ζ i, there are no more than [K : F(r1)] extensions of these isomor-

phisms, exactly [K : F(r1)] in case the roots of p̄(x) are distinct.Therefore, if the roots of p̄(x) are distinct, this has shown that there are

[K : F(r1)]m = [K : F(r1)] [F(r1) : F] = [K : F]

isomorphisms ofK to K̄ which agree with η on F. If the roots of p̄(x) are not distinct, thenmaybe there are fewer than [K : F] extensions of η .

Is this all of the isomorphisms? Suppose ζ is such an isomorphism of K and K̄. Thenconsider its restriction to F(r1) . By Corollary 9.4.6, this restriction must coincide withone of the ζ i chosen earlier. Then by induction, ζ is one of the extensions of the ζ i justmentioned. Thus, in particular, K and K̄ are isomorphic. ■

9.4.1 The Galois GroupFirst, here is the definition of a Group.

Definition 9.4.10 A group G is a nonempty set with an operation, denoted here as · suchthat the following axioms hold. (Often the operation is composition.)

1. For α,β ,γ ∈ G,(α ·β ) · γ = α · (β · γ) . We usually don’t bother to write the ·.

2. There exists ι ∈ G such that αι = ια = α

3. For every α ∈ G, there exists α−1 ∈ G such that αα−1 = α−1α = ι .

In Theorem 9.4.9, consider the case where F= F̄ and the isomorphism of F with itselfis just the identity.

Definition 9.4.11 WhenK is a finite extension of L, denote by G(K,L) the automorphismsof K which leave L fixed. For a finite set S, denote by |S| as the number of elements of S.

Most of the following theorem was shown earlier in Theorem 9.4.9.

Theorem 9.4.12 Let K be the splitting field of p(x) over the field F. Thus K consists ofF [a1, ...,an] where {a1, ...,an} are the roots of p(x). Then

|G(K,F)| ≤ [K : F] (9.22)