9.4. MORE ON ALGEBRAIC FIELD EXTENSIONS 215

This follows from the definition and the fact that g agrees with f on F. Indeed, if q(x) =amxm + · · ·+ a1x1 + a0, then, since g is an isomorphism which agrees with f on F, andg(r) = r̄,

g(q(r)) = f (am) r̄m + · · ·+ f (a1) r̄1 + f (a0)≡ q̄(r̄)

Therefore,

g◦α ([q(x)])≡ g(q(r)) = q̄(r̄) , ᾱ ◦h([q(x)]) = ᾱ ([ f (q(x))])≡ ᾱ ([q̄]) = q̄(r̄)

It follows that g = ᾱ ◦h◦α−1. Thus, there is only one such homomorphism and it is whatwas just obtained. ■

The following corollary emphasizes the main content of the above theorem.

Corollary 9.4.6 Let f : F→ F̄ be an isomorphism and let p̄(x) ≡ f (p(x)) where p(x)is the minimum polynomial for algebraic r. Then for each root r̄ of p̄(x) there is anisomorphism from F(r) to F̄(r̄) which extends f and satisfies g(r) = r̄. Also, if g is amonomophism extending f from F(r) to K̄ where K̄ is a field which contains all roots ofp̄(x) , then there is a root r̄ of p̄(x) such that g(r) = r̄.

Proof: It only remains to verify the last claim. Let g be a monomorphism. I need tofind a root r̄ of p̄(x). Let r̄ ≡ g(r) . Then if

p(x) = xm +am−1xm−1 + · · ·+a1x1 +a0,

it follows that

p̄(x) = xm + f (am−1)xm−1 + · · ·+ f (a1)x1 + f (a0)

= xm +g(am−1)xm−1 + · · ·+g(a1)x1 +g(a0)

and so,

p̄(g(r)) = g(r)m +g(am−1)g(r)m−1 + · · ·+g(a1)g(r)1 +g(a0)

= g(p(r)) = g(0) = 0 ■

Lemma 9.4.7 If p(x) is a monic irreducible polynomial, then it is the minimum polynomialfor each of its roots.

Proof: If r is a root of p(x) , then let q(x) be the minimum polynomial for r. Then

p(x) = q(x)k (x)+R(x)

where R(x) is 0 or else has smaller degree than q(x). However, R(r) = 0 and this contra-dicts q(x) being the minimum polynomial of r. Hence q(x) divides p(x) or else k (x) = 1.The latter possibility must be the case because p(x) is irreducible. ■

This lemma is about to be used in the proof of the following theorem. It involves thesplitting fieldsK= F [r1, ...,rm] ,K̄= F̄ [r̄1, ..., r̄m] of p(x) , p̄(x) where η is an isomorphismof F and F̄ as described above. See [26]. Here is a little diagram which describes what thistheorem says. It is about isomorphisms of K and K̄ which extend a given isomorphism η :F→ F̄.