Local Field - Non-archimedean Local Field Theory

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 : FZ ∪ {∞} 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:

    Eclecticism is the degree zero of contemporary general culture: one listens to reggae, watches a western, eats McDonald’s food for lunch and local cuisine for dinner, wears Paris perfume in Tokyo and “retro” clothes in Hong Kong; knowledge is a matter for TV games. It is easy to find a public for eclectic works.
    Jean François Lyotard (b. 1924)

    Yet, hermit and stoic as he was, he was really fond of sympathy, and threw himself heartily and childlike into the company of young people whom he loved, and whom he delighted to entertain, as he only could, with the varied and endless anecdotes of his experiences by field and river: and he was always ready to lead a huckleberry-party or a search for chestnuts and grapes.
    Ralph Waldo Emerson (1803–1882)

    The struggle for existence holds as much in the intellectual as in the physical world. A theory is a species of thinking, and its right to exist is coextensive with its power of resisting extinction by its rivals.
    Thomas Henry Huxley (1825–95)