Tsen's theorem

Remarks

For $K = k(t)$ the proof is a count of unknowns and equations. One may assume the coefficients of the form $f$ are polynomials in $t$ of degree at most $m$. One looks for a zero whose coordinates are polynomials in $t$ of degree at most $N$ with unknown coefficients. This gives $n(N + 1)$ unknowns in $k$, and $f = 0$ becomes $dN + m + 1$ homogeneous polynomial equations in them. Since $n > d$, for large $N$ there are more unknowns than equations, and over the algebraically closed field $k$ such a system has a nonzero solution. A finite extension of $k(t)$ is handled by a similar count, or by restriction of scalars.

A main consequence is that the Brauer group of $K$ is trivial: every central simple algebra over $K$ is a matrix algebra over $K$. For example, every quaternion algebra over $\C(t)$ splits.

The hypothesis that $k$ is algebraically closed is necessary. For $k = \R$, the form $x^2 + y^2 + z^2$ over $\R(t)$ has no nontrivial zero. Indeed, a nontrivial zero could be taken with polynomial coordinates, not all zero. Evaluating them at a real $t_0$ where they do not all vanish would give a nontrivial real zero of $x^2 + y^2 + z^2$.

Going deeper

For a student

Tsen's theorem is the basic example of a $C_1$ field besides finite fields and algebraically closed fields. In 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 $1$. Applied to quadratic forms in $3$ variables, it shows that every conic over such a field has a rational point. The analogous property for finite fields is the Chevalley–Warning theorem.