In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology. Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and those in which it is not. In the first case, one calls the local field an archimedean local field, in the second case, one calls it a non-archimedean local field. Local fields arise naturally in number theory as completions of global fields.
Every local field is isomorphic (as a topological field) to one of the following:
- Archimedean local fields (characteristic zero): the real numbers R, and the complex numbers C.
- Non-archimedean local fields of characteristic zero: finite extensions of the p-adic numbers Qp (where p is any prime number).
- Non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p).
There is an equivalent definition of non-archimedean local field: it is a field that is complete with respect to a discrete valuation and whose residue field is finite. However, some authors consider a more general notion, requiring only that the residue field be perfect, not necessarily finite. This article uses the former definition.
Read more about Local Field: Induced Absolute Value, Non-archimedean Local Field Theory, Higher Dimensional Local Fields
Famous quotes containing the words local and/or field:
“There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
—Adrienne Rich (b. 1929)
“Because mothers and daughters can affirm and enjoy their commonalities more readily, they are more likely to see how they might advance their individual interests in tandem, without one having to be sacrificed for the other.”
—Mary Field Belenky (20th century)