Quasi-algebraically closed field
Encyclopedia
In mathematics
, a field
F is called quasi-algebraically closed (or C1) if 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 variables
and of degree d satisfying
then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have
In geometric language, the hypersurface
defined by P, in projective space
of dimension N − 1, then has a point over F.
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
.
for k ≥ 1.
Lang and Nagata proved that if a field is Ck, then any extension of transcendence degree
n is Ck+n. The smallest k such that K is a Ck field (
if no such number exists), is called the diophantine dimension dd(K) of K.
Artin conjectured that p-adic fields were C2, 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 Qp with p large enough (depending on d).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a 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 variables
- X1, ..., XN,
and of degree d satisfying
- d < N
then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have
- P(x1, ..., xN) = 0.
In geometric language, the hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
defined by P, in projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of dimension N − 1, then has a point over F.
Examples and properties
- Any algebraically closed fieldAlgebraically closed fieldIn mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero. - Any finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
is quasi-algebraically closed by the Chevalley–Warning theorem. - Function fieldFunction fieldFunction field may refer to:*Function field of an algebraic variety*Function field...
s of algebraic curveAlgebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s over algebraically closed fields are quasi-algebraically closed by Tsen's theoremTsen's theoremIn mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed. This implies that the Brauer group of any such field vanishes, and more generally that all the Galois cohomology groups H i vanish for...
. - If K is a complete field with a discrete valuation and an algebraically closed residue field, then K is quasi-algebraically closed by a result of Lang.
- Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.
The Brauer group
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
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
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
in a 1936 paper; and later in the 1951 Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
dissertation of Serge Lang
Serge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
. The idea itself is attributed to Lang's advisor Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
.
Ck fields
Quasi-algebraically closed fields are also called C1. A Ck field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided- dk < N,
for k ≥ 1.
Lang and Nagata proved that if a field is Ck, then any extension of transcendence degree
Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...
n is Ck+n. The smallest k such that K is a Ck field (
if no such number exists), is called the diophantine dimension dd(K) of K.Artin conjectured that p-adic fields were C2, but
Guy Terjanian
Guy Terjanian
Guy Terjanian is a French-Armenian mathematician who has worked on algebraic number theory. He achieved his Ph.D under Claude Chevalley in 1966, and at that time published a counterexample to the original form of a conjecture of Emil Artin, which suitably modified had just been proved as the...
found p-adic counterexamples
Ax–Kochen theorem
The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic numbers in at least d2+1 variables has a nontrivial...
for all p. The Ax–Kochen theorem
Ax–Kochen theorem
The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic numbers in at least d2+1 variables has a nontrivial...
applied methods from model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
to show that Artin's conjecture was true for Qp with p large enough (depending on d).