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In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
topological field with respect to a non-discrete topology. Sometimes,
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
R, and the
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation 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 Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
as completions of
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global function fi ...
s. While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local fields, the fields of
p-adic number In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The exte ...
s for positive prime integer ''p'', were introduced by Kurt Hensel at the end of the 19th century. Every local field is isomorphic (as a topological field) to one of the following: *Archimedean local fields ( characteristic zero): the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
R, and the
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
C. *Non-Archimedean local fields of characteristic zero: finite extensions of the ''p''-adic numbers Q''p'' (where ''p'' is any
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
). *Non-Archimedean local fields of characteristic ''p'' (for ''p'' any given prime number): the field of formal Laurent series F''q''((''T'')) over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
F''q'', where ''q'' is a power of ''p''. In particular, of importance in number theory, classes of local fields show up as the completions of
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s with respect to their discrete valuation corresponding to one of their maximal ideals. Research papers in modern number theory often consider a more general notion, requiring only that the residue field be
perfect Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 f ...
of positive characteristic, not necessarily finite. This article uses the former definition.


Induced absolute value

Given such an absolute value on a field ''K'', the following topology can be defined on ''K'': for a positive real number ''m'', define the subset ''B''m of ''K'' by :B_m:=\. Then, the ''b+B''m make up a neighbourhood basis of b in ''K''. Conversely, a topological field with a non-discrete locally compact topology has an absolute value defining its topology. It can be constructed using the Haar measure of the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
of the field.


Basic features of non-Archimedean local fields

For a non-Archimedean local field ''F'' (with absolute value denoted by , ·, ), the following objects are important: *its
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
\mathcal = \ which is a
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R' ...
, is the closed
unit ball Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
of ''F'', and is compact; *the units in its ring of integers \mathcal^\times = \ which forms a group and is the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A u ...
of ''F''; *the unique non-zero prime ideal \mathfrak in its ring of integers which is its open unit ball \; *a generator \varpi of \mathfrak called a uniformizer of F; *its residue field k=\mathcal/\mathfrak which is finite (since it is compact and discrete). Every non-zero element ''a'' of ''F'' can be written as ''a'' = ϖ''n''''u'' with ''u'' a unit, and ''n'' a unique integer. The normalized valuation of ''F'' is the
surjective function In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
''v'' : ''F'' → Z ∪ defined by sending a non-zero ''a'' to the unique integer ''n'' such that ''a'' = ϖ''n''''u'' with ''u'' a unit, and by sending 0 to ∞. If ''q'' is the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the residue field, the absolute value on ''F'' induced by its structure as a local field is given by: :, a, =q^. An equivalent and very important 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.


Examples

#The ''p''-adic numbers: the ring of integers of Q''p'' is the ring of ''p''-adic integers Z''p''. Its prime ideal is ''p''Z''p'' and its residue field is Z/''p''Z. Every non-zero element of Qp can be written as ''u'' ''p''''n'' where ''u'' is a unit in Z''p'' and ''n'' is an integer, then ''v''(''u'' ''p''n) = ''n'' for the normalized valuation. #The formal Laurent series over a finite field: the ring of integers of F''q''((''T'')) is the ring of formal power series F''q'' ''T''. Its maximal ideal is (''T'') (i.e. the power series whose constant term is zero) and its residue field is F''q''. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows: #::v\left(\sum_^\infty a_iT^i\right) = -m (where ''a''−''m'' is non-zero). #The formal Laurent series over the complex numbers is ''not'' a local field. For example, its residue field is C ''T''/(''T'') = C, which is not finite.


Higher unit groups

The ''n''th higher unit group of a non-Archimedean local field ''F'' is :U^=1+\mathfrak^n=\left\ for ''n'' ≥ 1. The group ''U''(1) is called the group of principal units, and any element of it is called a principal unit. The full unit group \mathcal^\times is denoted ''U''(0). The higher unit groups form a decreasing
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
of the unit group :\mathcal^\times\supseteq U^\supseteq U^\supseteq\cdots whose quotients are given by :\mathcal^\times/U^\cong\left(\mathcal/\mathfrak^n\right)^\times\text\,U^/U^\approx\mathcal/\mathfrak for ''n'' ≥ 1. (Here "\approx" means a non-canonical isomorphism.)


Structure of the unit group

The multiplicative group of non-zero elements of a non-Archimedean local field ''F'' is isomorphic to :F^\times\cong(\varpi)\times\mu_\times U^ where ''q'' is the order of the residue field, and μ''q''−1 is the group of (''q''−1)st roots of unity (in ''F''). Its structure as an abelian group depends on its characteristic: *If ''F'' has positive characteristic ''p'', then ::F^\times\cong\mathbf\oplus\mathbf/\oplus\mathbf_p^\mathbf :where N denotes the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s; *If ''F'' has characteristic zero (i.e. it is a finite extension of Q''p'' of degree ''d''), then ::F^\times\cong\mathbf\oplus\mathbf/(q-1)\oplus\mathbf/p^a\oplus\mathbf_p^d :where ''a'' ≥ 0 is defined so that the group of ''p''-power roots of unity in ''F'' is \mu_.


Theory of local fields

This theory includes the study of types of local fields, extensions of local fields using Hensel's lemma, Galois extensions of local fields, ramification groups filtrations of
Galois groups In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of local fields, the behavior of the norm map on local fields, the local reciprocity homomorphism and existence theorem in local class field theory, local Langlands correspondence, Hodge-Tate theory (also called
p-adic Hodge theory In mathematics, ''p''-adic Hodge theory is a theory that provides a way to classify and study ''p''-adic Galois representations of characteristic 0 local fields with residual characteristic ''p'' (such as Q''p''). The theory has its beginnings ...
), explicit formulas for the Hilbert symbol in local class field theory, see e.g.


Higher-dimensional local fields

A local field is sometimes called a ''one-dimensional local field''. A non-Archimedean local field can be viewed 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 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 perfect field of positive characteristic. 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.


See also

* Hensel's lemma * Ramification group * Local class field theory * Higher local field


Citations


References

* * * * *


External links

* {{DEFAULTSORT:Local Field Field (mathematics) Algebraic number theory