Difference between revisions of "Quasi-algebraically closed field"
From Encyclopedia of Mathematics
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| − | A field $K$ for which every homogeneous polynomial form over $K$ of degree $d$ in $n$ | + | 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 $d | + | 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. |
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| + | See also: [[Pseudo algebraically closed field]]. | ||
====References==== | ====References==== | ||
| − | * Lang, Serge ''Survey of diophantine geometry'' Springer (1997) ISBN 3-540-61223-8 {{ZBL|0869.11051}} | + | * 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}} | + | * Lorenz, Falko ''Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics'' Springer (2008) {{ISBN|978-0-387-72487-4}} {{ZBL|1130.12001}} |
Latest revision as of 19:37, 17 November 2023
$C_1$ 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=37144
Quasi-algebraically closed field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-algebraically_closed_field&oldid=37144