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Difference between revisions of "Quasi-algebraically closed field"

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(See also Pseudo algebraically closed field)
(→‎References: isbn link)
 
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====References====
 
====References====
* Lang, Serge ''Survey of diophantine geometry'' Springer (1997) ISBN 3-540-61223-8 {{ZBL|0869.11051}}
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* 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}}
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* 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=37150