Non-archimedean Local Field Theory
For a non-archimedean local field F (with absolute value denoted by |·|), the following objects are important:
- its ring of integers which is a discrete valuation ring, is the closed unit ball of F, and is compact;
- the units in its ring of integers which forms a group and is the unit sphere of F;
- the unique non-zero prime ideal in its ring of integers which is its open unit ball ;
- a generator ϖ of called a uniformizer of F;
- its residue field which is finite (since it is compact and discrete).
Every non-zero element a of F can be written as a = ϖnu with u a unit, and n a unique integer. The normalized valuation of F is the surjective function v : F → Z ∪ {∞} defined by sending a non-zero a to the unique integer n such that a = ϖnu with u a unit, and by sending 0 to ∞. If q is the cardinality of the residue field, the absolute value on F induced by its structure as a local field is given by
An equivalent definition of a non-archimedean local field is that it is a field that is complete with respect to a discrete valuation and whose residue field is finite.
Read more about this topic: Local Field
Famous quotes containing the words local, field and/or theory:
“While it may not heighten our sympathy, wit widens our horizons by its flashes, revealing remote hidden affiliations and drawing laughter from far afield; humor, in contrast, strikes up fellow feeling, and though it does not leap so much across time and space, enriches our insight into the universal in familiar things, lending it a local habitation and a name.”
—Marie Collins Swabey. Comic Laughter, ch. 5, Yale University Press (1961)
“They talk about a woman’s sphere,
As though it had a limit.
There’s not a place in earth or heaven.
There’s not a task to mankind given ...
Without a woman in it.”
—Kate Field (1838–1896)
“The theory [before the twentieth century] ... was that all the jobs in the world belonged by right to men, and that only men were by nature entitled to wages. If a woman earned money, outside domestic service, it was because some misfortune had deprived her of masculine protection.”
—Rheta Childe Dorr (1866–1948)