Local field - Citizendia

In mathematics, a local field is a special type of field that has a non-trivial absolute value and which is a locally compact topological field with respect to this absolute value. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an absolute value is a function which measures the "size" of elements in a field or Integral domain. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are There are two basic types of local field: those in which the absolute value is Archimedean and those in which it is non-Archimedean. 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. There is an equivalent definition of non-archimedean local field given below. Local fields arise naturally in number theory as completions of global fields. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, the term global field refers to either of the following a number field, i

The complete classification of local fields (up to isomorphism) is the following:

Archimedean local fields (characteristic zero): the real numbers R, and the complex numbers C. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
Non-archimedean local fields of characteristic zero: finite extensions of the p-adic numbers Q_p . In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897
Non-archimedean local fields of characteristic p: finite extensions of the field of formal Laurent series F_q((T)) over a finite field F_q. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements

1 Non-Archimedean local fields
2 Examples
3 See also
4 References

Non-Archimedean local fields

For a non-archimedean local field $F$ , the following objects are very important:

its ring of integers $\mathcal{O}$ which is its closed unit ball $\{a\in F: |a|\leq 1\}$ (it is compact),
the units in its ring of integers $\mathcal{O}^\times$ which is its unit sphere $\{a\in F: |a|= 1\}$ ,
the unique prime ideal in its ring of integers $\mathfrak{m}$ which is its open unit ball $\{a\in F: |a|< 1\}$ ,
its residue field $k=\mathcal{O}/\mathfrak{m}$ which is finite (since it is compact and discrete). In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated "

One often talks about the (discrete) valuation of a non-archimedean local field. Valuation in mathematics may refer to Valuation (algebra Valuation (logic Valuation (measure theory This is a map $v:F\rightarrow\mathbb{R}\cup\{\infty\}$ obtained as follows: there is a real number 0 < c < 1 such that

$c^{v(a)}=|a|\mbox{ for all }a\in F$ .

One generally chooses c such that v surjects onto $\mathbb{Z}\cup\{\infty\}$ , and calls this the normalized valuation.

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.

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 pZ_p and its residue field is Z/pZ. Every non-zero element of Q_p can be written as u pⁿ where u is a unit in Z_p and n is an integer, then v(u pⁿ) = 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]]. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not Its prime ideal is (T) (i. e. the power series whose constant term is zero) and its residue field is F_q. In Mathematics, the constant term of a Polynomial is the term of degree 0 Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows: $v\left(\sum_{i=-m}^\infty a_iT^i\right) = -m$ (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.

References

Milne, James, Algebraic Number Theory. In Mathematics, Helmut Hasse 's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation
Serre, Jean-Pierre (1995). Local Fields. Springer-Verlag. ISBN 0-387-90424-7.

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Contents

Non-Archimedean local fields

Examples

See also

References