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:
“The higher the climb, the heavier the fall.”
—Chinese proverb.
“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)
“Surely there must be some way to find a husband or, for that matter, merely an escort, without sacrificing one’s privacy, self-respect, and interior decorating scheme. For example, men could be imported from the developing countries, many parts of which are suffering from a man excess, at least in relation to local food supply.”
—Barbara Ehrenreich (b. 1941)
“Nature will not let us fret and fume. She does not like our benevolence or our learning much better than she likes our frauds and wars. When we come out of the caucus, or the bank, or the abolition-convention, or the temperance-meeting, or the transcendental club, into the fields and woods, she says to us, “so hot? my little Sir.””
—Ralph Waldo Emerson (1803–1882)