Quasi-algebraically closed field

Summary

A field is quasi-algebraically closed, or $C_1$, if every homogeneous polynomial with more variables than its degree has a zero other than the zero vector. Every algebraically closed field is $C_1$, and so is every finite field, by the Chevalley–Warning theorem. By Tsen's theorem, the function field of a curve over an algebraically closed field is $C_1$. The Brauer group of a $C_1$ field is trivial, which is one reason the notion matters.