Rationality of some genus $0$ extensions of $K(X)$

Abstract:

Let $L$ be a quadratic extension of a field $K$ with Galois group $\left \{ {e,\Gamma } \right \}$. Let $\left \{ {x,y} \right \}$ be algebraically independent over $L$ and let $\Gamma$ be extended to an automorphism of $L(x,y)$ so that $\Gamma (x) = x$ and the extension has order 2. Then $L{(x,y)^\Gamma }$ is a genus 0 extension of $K(x)$. This paper examines when $L{(x,y)^\Gamma }$ will be pure transcendental over $K$. It is shown that some important examples from field theory can be realized by this construction. The main result shows that $L{(x,y)^\Gamma }$ is pure transcendental over $K$ if $\Gamma (y) = ({x^2} + bx + c)/y(\operatorname {char} K \ne 2)$. An example illustrates that it is essential that the second degree polynomial be monic.