Let / g K(x) with / g K and ,g relatively prime in K[x] and

Let ∫/ g ϵ K(x) with ∫/ g ∉ K and ∫,g relatively prime in K[x] and consider the extension of K by K(x). 

(a) x is algebraic over K(∫/g) and [K(x): K(∫/g)] = max (deg ∫,deg g).

(b) If E ≠ K is an intermediate field, then [K(x) : E] is finite.

(c) The assignment x |→ ∫/g induces a homomorphism σ : K(x) → K(x) such that φ(x)/ψ(x)|φ(∫/g)/ψ(∫/g). σ is a K automorphism of K(x) if and only if max (deg ∫,deg g) = 1.

(d) Aut_KK(x) consists of all those automorphisms induced (as in (c)) by the assignment 

where a,b,c,d e Kand ad - be ¢. 0. 

Transcribed Image Text:

x (ax + b)/(cx + d),