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:
“[Urging the national government] to eradicate local prejudices and mistaken rivalships to consolidate the affairs of the states into one harmonious interest.”
—James Madison (17511836)
“And they wonder, as waiting the long years through
In the dust of that little chair,
What has become of our Little Boy Blue,
Since he kissed them and put them there.”
—Eugene Field (18501895)
“The things that will destroy America are prosperity-at-any- price, peace-at-any-price, safety-first instead of duty-first, the love of soft living, and the get-rich-quick theory of life.”
—Theodore Roosevelt (18581919)