View source for 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}}