Skip to main content

Mathematics (thesaurus)

Search from vocabulary

align
annotate
download

Concept information

geometry > algebraic geometry > Diophantine geometry > quasi-algebraically closed field
number > number theory > Diophantine geometry > quasi-algebraically closed field
algebra > abstract algebra > algebraic structure > field > quasi-algebraically closed field

Preferred term

quasi-algebraically closed field  

Definition

  • 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. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper (Tsen 1936); and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper (Lang 1952). The idea itself is attributed to Lang's advisor Emil Artin. Formally, 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 defined by P, in projective space of degree N − 2, then has a point over F.
    (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Quasi-algebraically_closed_field)

Broader concept

Related concepts

In other languages

URI

http://data.loterre.fr/ark:/67375/PSR-KTBMST62-P

Download this concept:

RDF/XML TURTLE JSON-LD Created 8/24/23, last modified 10/18/24