- Quasi-algebraically closed field
In

mathematics , a field "F" is called**quasi-algebraically closed**(or**C**) if for every non-constant_{1}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

indeterminate s:"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", inprojective 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 field s ofalgebraic curve s 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 ofEmmy Noether in a 1936 paper; and later in the 1951Princeton University dissertation ofSerge Lang . The idea itself is attributed to Lang's advisorEmil Artin .**"C"**_{k}fieldsQuasi-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" ≥ 1.

Lang and Nagata proved that if a field is "C"

_{"k"}, then any extension oftranscendence 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". TheAx-Kochen theorem applied methods frommodel 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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