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:
“With two sons born eighteen months apart, I operated mainly on automatic pilot through the ceaseless activity of their early childhood. I remember opening the refrigerator late one night and finding a roll of aluminum foil next to a pair of small red tennies. Certain that I was responsible for the refrigerated shoes, I quickly closed the door and ran upstairs to make sure I had put the babies in their cribs instead of the linen closet.”
—Mary Kay Blakely (20th century)
“In the field of world policy I would dedicate this Nation to the policy of the Good Neighbor—the neighbor who resolutely respects himself and, because he does, respects the rights of others—the neighbor who respects his obligations and respects the sanctity of his agreements in and with a world of neighbors.”
—Franklin D. Roosevelt (1882–1945)