Rationality problem of generalized Ch\^atelet surfaces

The surface $z^2=ay^2+P(x), \, a \in k, \, P(x) \in k[x]$ is not $k$-rational, if $a \not\in k^2$ and $P(x)$ satisfies some conditions. This result essentially due to Iskovskih but his statement is in terms of algebraic geometry, and not so easy to access for the researchers of the field extension. This paper aims to give a formulation accessible more easily. A necessary and sufficient condition for $k$-rationality is given in terms of $a$ and $P$, assuming that $\ch k \not= 2$ and every irreducible component of $P(x)$ is separable over $k$.