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  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ be an integral [[algebraic variety|HV6YBS0YDH]] over $k$ of dimension $n$.\nThe variety $X$ is *rational* over $k$ if it is birationally equivalent to the [[projective space|ZT9A6JK8QF]] $\\mathbb{P}^n_k$.\nThis means that there is a [[birational map|KA5890R9YM]] $\\mathbb{P}^n_k \\dashrightarrow X$.\n\n## Remarks\n\nEquivalently, the [[function field|2D5SS6WM14]] $k(X)$ 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|D7TR0YWR9H]] $n$.\nExample: the conic $C \\colon x^2 + y^2 = z^2$ in $\\mathbb{P}^2_\\Q$ is rational.\nThe map $[s : t] \\mapsto [s^2 - t^2 : 2st : s^2 + t^2]$ sends $\\mathbb{P}^1$ into $C$, since $(s^2 - t^2)^2 + (2st)^2 = (s^2 + t^2)^2$.\nIts inverse is the projection $[x : y : z] \\mapsto [y : z - x]$ from the point $[1 : 0 : 1]$ of $C$.\nNon-example: the conic $x^2 + y^2 + z^2 = 0$ in $\\mathbb{P}^2_\\R$ is not rational over $\\R$.\nIt has no real points, while a variety birational to $\\mathbb{P}^1_\\R$ has a dense set of real points.\nOver $\\C$ the same conic is rational.\nSo rationality depends on the field $k$.\n",
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