# 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

**How to Cite This Entry:**

Quasi-algebraically closed field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Quasi-algebraically_closed_field&oldid=37150