Local field in the context of Complete metric space

In mathematics, certain subsets of some fields are called orders. The set of integers is an order in the rational numbers (the only one). In an algebraic number field ⁠${\displaystyle K}$⁠, an order is a ring of algebraic integers whose field of fractions is ⁠${\displaystyle K}$⁠, and the maximal order, often denoted ⁠${\displaystyle {\mathcal {O}}_{K}}$⁠, is the ring of all algebraic integers in ⁠${\displaystyle K}$⁠. In a non-Archimedean local field ⁠${\displaystyle K}$⁠, an order is a subring which is generated by finitely many elements of non-negative valuation. In that case, the maximal order, denoted ⁠${\displaystyle {\mathcal {O}}_{K}}$⁠, is the valuation ring formed by all elements of non-negative valuation.

Giving the same name to such seemingly different notions is motivated by the local–global principle that relates properties of a number field with properties of all its local fields.

In mathematics, a Galois module is a G-module, with G being the Galois group (named for Évariste Galois) of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field.

For an abelian variety ${\displaystyle A}$ defined over a field ${\displaystyle F}$ with ring of integers ${\displaystyle R}$, consider the Néron model of ${\displaystyle A}$, which is a 'best possible' model of ${\displaystyle A}$ defined over ${\displaystyle R}$. This model may be represented as a scheme over ${\displaystyle \mathrm {Spec} (R)}$ (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism${\displaystyle \mathrm {Spec} (F)\to \mathrm {Spec} (R)}$gives back ${\displaystyle A}$. The Néron model is a smooth group scheme, so we can consider ${\displaystyle A^{0}}$, the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a residue field ${\displaystyle k}$, ${\displaystyle A_{k}^{0}}$ is a group variety over ${\displaystyle k}$, hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that ${\displaystyle A_{k}^{0}}$ is a semiabelian variety, then ${\displaystyle A}$ has semistable reduction at the prime corresponding to ${\displaystyle k}$. If ${\displaystyle F}$ is a global field, then ${\displaystyle A}$ is semistable if it has good or semistable reduction at all primes.

In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.

Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem).

In mathematics, local class field theory (LCFT), introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Q_p (where p is any prime number), or the field of formal Laurent series F_q((T)) over a finite field F_q.