2.1. Maps from simple extensions.

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Let E and E_ be fields containing F. An F-homomorphism is a homomorphism ϕ :

E → E_ such that ϕ(a) = a for all a ∈ F. Thus an F-h _ omorphism maps a polynomial

ai1···imαi1

1

・ ・ ・ αim

m , ai1···im

∈ F, to

_

ai1···imϕ(α1)i1 ・ ・ ・ ϕ(αm)im.

An F-isomorphism is a bijective F-homomorphism. Note that if E and E_ have the same

finite degree over F, then an F-homomorphism is automatically an F-isomorphism.

Proposition 2.1. Let F(α) be a simple field extension of a field F, and let Ω be a second

field containing F.

(a) Assume α is transcendental over F; then for any F-homomorphism ϕ : F(α) → Ω, ϕ(α)

is transcendental over F, and the map ϕ → ϕ(α) defines a one-to-one correspondence

{F-homomorphisms ϕ : F(α) → Ω} ↔ { elements of Ω transcendental over F}.

(b) Assume α is algebraic over F, with minimum polynomial f(X); then for any Fhomomorphism

ϕ : F[α] → Ω, ϕ(α) is a root of f(X) in Ω, and the map ϕ → ϕ(α)

defines a one-to-one correspondence

{F-homomorphisms ϕ : F[α]→ Ω} ↔ { distinct roots of f(X) in Ω}.

In particular, the number of such maps is the number of distinct roots of f in Ω.

Proof. (a) Let γ ∈ Ω. To say that α is transcendental over F means that F[α] is the

ring of polynomials in α (as variable). By the universal property of polynomial rings, there

is a unique F-homomorphism ϕ : F[α] → Ω sending α to γ. This extends to F(α) if and

only if all nonzero elements of F[α] are sent to invertible (i.e., nonzero) elements of Ω, which

is so if and only if γ is transcendental.

(b) Let f(X) =

_

aiXi, and consider an F-homomorphism ϕ : F[α] → Ω. On applying

ϕ to the equation

_

aiαi = 0, we obtain the equation

_

aiϕ(α)i = 0, which shows that

γ =df ϕ(α) is a root of f(X) in Ω. Conversely, let γ ∈ Ω be a root of f(X). The map

F[X] → Ω, g(X) → g(γ), factors through F[X]/(f(X)). When composed with the inverse

of the isomorphism F[X]/(f(X)) → F[α], it becomes a homomorphism F[α] → Ω sending

α to γ.

We shall need a slight generalization of this result.

Proposition 2.2. Let F(α) be a simple field extension of a field F, and let ϕ0 : F → Ω

be a homomorphism of F into a second field Ω.

(a) Assume α is transcendental over F; then the map ϕ → ϕ(α) defines a one-to-one

correspondence

{extensions ϕ : F(α) → Ω of ϕ0} ↔ {elements of Ω transcendental over ϕ0(F)}.

(b) Assume α is algebraic over F, with minimum polynomial f(X); then the map ϕ → ϕ(α)

defines a one-to-one correspondence

{extensions ϕ : F[α]→ Ω of ϕ0} ↔ { distinct roots of (ϕ0f)(X)in Ω}.

In particular, the number of such maps is the number of distinct roots of ϕ0f in Ω.

FIELDS AND GALOIS THEORY 13

Proof. The proof is essentially the same as that of the preceding proposition.

By ϕ0f we mean the polynomial obtained by applying ϕ0 to the coefficients of f, i.e.,

f =

_

aiXi =⇒ ϕ0f =

_

ϕ(ai)Xi.

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