Quasi-algebraically Closed Field
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; and later in the 1951 Princeton University dissertation of Serge Lang. 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 dimension N − 1, then has a point over F.
Read more about Quasi-algebraically Closed Field: Examples, Properties, Ck Fields
Famous quotes containing the words closed and/or field:
“Because you live, O Christ,
the spirit bird of hope is freed for flying,
our cages of despair no longer keep us closed and life-denying.
The stone has rolled away and death cannot imprison!
O sing this Easter Day, for Jesus Christ has risen!”
—Shirley Erena Murray (20th century)
“It is through attentive love, the ability to ask “What are you going through?” and the ability to hear the answer that the reality of the child is both created and respected.”
—Mary Field Belenky (20th century)