Rational variety

Remarks

Equivalently, the function field $k(X)$ is purely transcendental over $k$: it is isomorphic to $k(t_1, \ldots, t_n)$ for elements $t_1, \ldots, t_n$ of transcendence degree $n$. Example: the conic $C \colon x^2 + y^2 = z^2$ in $\mathbb{P}^2_\Q$ is rational. The map $[s : t] \mapsto [s^2 - t^2 : 2st : s^2 + t^2]$ sends $\mathbb{P}^1$ into $C$, since $(s^2 - t^2)^2 + (2st)^2 = (s^2 + t^2)^2$. Its inverse is the projection $[x : y : z] \mapsto [y : z - x]$ from the point $[1 : 0 : 1]$ of $C$. Non-example: the conic $x^2 + y^2 + z^2 = 0$ in $\mathbb{P}^2_\R$ is not rational over $\R$. It has no real points, while a variety birational to $\mathbb{P}^1_\R$ has a dense set of real points. Over $\C$ the same conic is rational. So rationality depends on the field $k$.