Higher Dimensional Local Fields
It is natural to introduce non-archimedean local fields in a uniform geometric way as the field of fractions of the completion of the local ring of a one-dimensional arithmetic scheme of rank 1 at its non-singular point. For generalizations, a local field is sometimes called a one-dimensional local field.
For a non-negative integer n, an n-dimensional local field is a complete discrete valuation field whose residue field is an (n − 1)-dimensional local field. Depending on the definition of local field, a zero-dimensional local field is then either a finite field (with the definition used in this article), or a quasi-finite field, or a perfect field.
From the geometric point of view, n-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an n-dimensional arithmetic scheme.
Read more about this topic: Local Field
Famous quotes containing the words higher, dimensional, local and/or fields:
“Wherever a man separates from the multitude, and goes his own way in this mood, there indeed is a fork in the road, though ordinary travelers may see only a gap in the paling. His solitary path across lots will turn out the higher way of the two.”
—Henry David Thoreau (1817–1862)
“I don’t see black people as victims even though we are exploited. Victims are flat, one- dimensional characters, someone rolled over by a steamroller so you have a cardboard person. We are far more resilient and more rounded than that. I will go on showing there’s more to us than our being victimized. Victims are dead.”
—Kristin Hunter (b. 1931)
“This is the only “wet” community in a wide area, and is the rendezvous of cow hands seeking to break the monotony of chuck wagon food and range life. Friday night is the “big time” for local cowboys, and consequently the calaboose is called the “Friday night jail.””
—Administration in the State of Texa, U.S. public relief program (1935-1943)
“O do not grieve
for a torn earth
barren fields burnt forests
cracked riven volcano-broken
hill-slopes islands shrunken
mountains lost.”
—Hilda Doolittle (1886–1961)