Quasi-algebraically closed field

$C_1$ field

A field $K$ for which every homogeneous polynomial form over $K$ of degree $d$ in $n$ variables with $n > d$ has a non-trivial zero in $K$. Clearly every algebraically closed field is quasi-algebraically closed. Further examples are given by function fields in one variable over algebraically closed fields: this is Tsen's theorem. Chevalley proved that finite fields are QAC. A finite extension of a QAC field is again QAC. The Brauer group of a QAC field is trivial.

A fields is strongly quasi-algebraically closed if the same properties holds for all polynomial forms. More generally, a field is $C_i$ if every form with $n > d^i$ has a non-trivial zero.

See also: Pseudo algebraically closed field.

References

• Lang, Serge Survey of diophantine geometry Springer (1997) ISBN 3-540-61223-8 Zbl 0869.11051
• Lorenz, Falko Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics Springer (2008) ISBN 978-0-387-72487-4 Zbl 1130.12001