Exercise 11, § 3

Let E be an extension of a field K, $x \in E$ a transcendental element over K, so that $K(x)$ is isomorphic to the field $K(T)$ of rational fractions in an indeterminate over K. Thus every $y \in K(x)$ may be written $y = g(x)/h(x)$, where g and h are two polynomials of $K[X]$ that are relatively prime (IV, p. 12), determined up to a non-zero common factor in K; the larger of the degrees of g and h is called the height of y with respect to x. a) Show that in the polynomial ring $K(y)[X]$ the polynomial $g(X) - yh(X)$ is irreducible if $y \notin K$. Deduce that if y has height $n > 0$ with respect to x, then $K(x)$ is an algebraic extension of degree $n$ of $K(y)$.

Let $P(X) = \sum_{i=0}^{n} y_i X^i$ be the minimal polynomial of $x$ over $K(y)$. Show that for $0 \leq i \leq n$ we have either $y_i \in K$ or $K(y_i) = K(y)$.

b) Deduce from a) that every $y \in K(x)$ such that $K(y) = K(x)$ is of the form $(ax + b)/(cx + d)$ where $a, b, c, d$ are elements of $K$ such that $ad - bc \neq 0$; converse. Find all $K$-automorphisms of $K(x)$.

c) Show that if $y \in K(x)$ is of height $n$ with respect to $x$, and $z \in K(y)$ of height $m$ with respect to $y$, then $z$ is of height $mn$ with respect to $x$.

\* d) Let $F$ be an extension of $K$ such that $K \subset F \subset K(x)$ and $F \neq K$; let $y$ be an element of $F$ whose height $m$ with respect to $x$ has the least possible value; show that $F = K(y)$ ("Liuroth's theorem"). (Let $P$ be the minimal polynomial of $x$ in $F[T]$; show that $P(T) = Q(T, x)/R(x)$, where $Q$ is a polynomial of $K[T, X]$ of degree $\geq m$ in $X$ and is not divisible by any non-constant polynomial of $K[X]$, and $R$ is a polynomial of $K[X]$. If $y = g(x)/h(x)$, where $g$ and $h$ are relatively prime in $K[X]$, observe that by a) the polynomial $g(T)h(X) - g(X)h(T)$ is not divisible by any non-constant polynomial of $K[T]$ or of $K[X]$; deduce that necessarily

where $c \in K$,

using Ex. 10.)