Exercise 18, § 2

Let K be a commutative field and K[X] the ring of polynomials in one indeterminate over K. Show that every automorphism s of the ring K[X] leaves K invariant (Exercise 14(c)) and induces on K an automorphism of this field; show further that necessarily s(X) = aX + b, with a ≠ 0 in K and $b \in K$ (cf. Exercise 17). Conversely, show that being given an automorphism $\sigma$ of $K$ and two elements $a, b$ of $K$ such that $a \neq 0$ defines one and only one automorphism $s$ of $K[X]$ such that $s|K = \sigma$ and $s(X) = aX + b$. If $G$ is the group of all automorphisms of the ring $K[X]$ and $N$ the subgroup of $G$ consisting of the $K$-algebra structure of $K[X]$, show that $N$ is a normal subgroup of $G$ and that $G/N$ is isomorphic to the automorphism group of the field $K$; the group $N$ is isomorphic to the group defined on the set $K^* \times K$ by the law of composition $(\lambda, \mu)(\lambda', \mu') = (\lambda \lambda', \lambda' \mu + \mu')$.