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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|02SJG1HC32]].\nLet $K$ be a [[function field|XKEXQDRTGH]] of one variable over $k$, that is, a finitely generated field extension of $k$ of [[transcendence degree|D7TR0YWR9H]] $1$.\nThen $K$ is [[quasi-algebraically closed|D637VGM7GV]].\nThat is, every [[homogeneous polynomial|SD3M01318E]] over $K$ of degree $d \\ge 1$ in $n \u003e d$ variables has a nonzero zero in $K^n$.\n\n## Remarks\n\nFor $K = k(t)$ the proof is a count of unknowns and equations.\nOne may assume the coefficients of the form $f$ are polynomials in $t$ of degree at most $m$.\nOne looks for a zero whose coordinates are polynomials in $t$ of degree at most $N$ with unknown coefficients.\nThis gives $n(N + 1)$ unknowns in $k$, and $f = 0$ becomes $dN + m + 1$ homogeneous polynomial equations in them.\nSince $n \u003e d$, for large $N$ there are more unknowns than equations, and over the algebraically closed field $k$ such a system has a nonzero solution.\nA finite extension of $k(t)$ is handled by a similar count, or by restriction of scalars.\n\nA main consequence is that the [[Brauer group|BEW48TA9WZ]] of $K$ is trivial: every central simple algebra over $K$ is a matrix algebra over $K$.\nFor example, every quaternion algebra over $\\C(t)$ splits.\n\nThe hypothesis that $k$ is algebraically closed is necessary.\nFor $k = \\R$, the form $x^2 + y^2 + z^2$ over $\\R(t)$ has no nontrivial zero.\nIndeed, a nontrivial zero could be taken with polynomial coordinates, not all zero.\nEvaluating them at a real $t_0$ where they do not all vanish would give a nontrivial real zero of $x^2 + y^2 + z^2$.\n\n## Explanations\n\n### L1 human\n\nOver $\\C$, a polynomial in one variable always has a root.\nTsen's theorem says that the field $\\C(t)$ of rational functions is only slightly less closed.\nA form over $\\C(t)$ with more variables than its degree always has a nonzero solution.\nFor example, the quadratic form $x^2 + t y^2 + (t + 1) z^2$ has a nonzero zero over $\\C(t)$.\n\n### L2 human\n\nTsen's theorem is the basic example of a $C_1$ field besides finite fields and algebraically closed fields.\nIn arithmetic geometry it says that the function field of a curve over an algebraically closed field behaves like a field of cohomological dimension at most $1$.\nApplied to quadratic forms in $3$ variables, it shows that every conic over such a field has a rational point.\nThe analogous property for finite fields is the [[Chevalley–Warning theorem|08895Q5W96]].\n",
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