Quasi-algebraically closed field

Quasi-algebraically closed field

In mathematics, a field "F" is called quasi-algebraically closed (or C1) if for every non-constant homogeneous polynomial "P" over "F" has a non-trivial zero provided the number of its variables is more than its degree.

In other words, if "P" is a non-constant homogeneous polynomial in indeterminates

:"X"1, ..., "X""N",

and of degree "d" satisfying

:"d" < "N"

then it has a non-trivial zero over "F"; that is, for some "x""i" in "F", not all 0, we have

:"P"("x""1", ..., "x""N") = 0.

In geometric language, the hypersurface defined by "P", in projective space of dimension "N" − 1, then has a point over "F".

Examples and properties

*Any finite field is quasi-algebraically closed. (Chevalley-Warning theorem)
*Function fields of algebraic curves over algebraically closed fields are quasi-algebraically closed.(Tsen's theorem).
*Lang showed that if "K" is a complete field with a discrete valuation and an algebraically closed residue field, then "K" is quasi-algebraically closed.
*Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.

The Brauer group of a quasi-algebraically closed field is trivial.

The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether in a 1936 paper; and later in the 1951 Princeton University dissertation of Serge Lang. The idea itself is attributed to Lang's advisor Emil Artin.

"C"k fields

Quasi-algebraically closed fields are also called "C"1. A "C"k field, more generally, is one for which any homogeneous polynomial of degree "d" in "N" variables has a non-trivial zero, provided

:"d""k" < "N",

for "k" &ge; 1.

Lang and Nagata proved that if a field is "C""k", then any extension of transcendence degree "n" is "C""k"+"n".

Artin conjectured that "p"-adic fields were "C"2, but Guy Terjanian found "p"-adic counterexamples for all "p". The Ax-Kochen theorem applied methods from model theory to show that Artin's conjecture was true for Q"p" with "p" large enough (depending on "d").

References

*C. Tsen, "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer K"orper", J. Chinese Math. Soc. 171 (1936), 81-92
*Serge Lang, "On quasi algebraic closure", Annals of Mathematics 55 (1952), 373–390.
*M. J. Greenberg , "Lectures on Forms in Many Variables", Benjamin, 1969.
*J. Ax and S. Kochen. "Diophantine problems over local fields I" Amer. J. Math., 87:605-630, 1965
*J.-P. Serre, "Galois cohomology", ISBN 3-540-61990-9


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