In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. There is some inconsistency in usage, but usually a local field is further assumed to be locally compact, and often the field of real numbers and the field of complex numbers are considered to be local as well by virtue of their local compactness. We insist on local compactness but exclude R and C in the discussion below. Local fields arise naturally in number theory as completions of global fields, especially number fields. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a discrete valuation on a commutative ring A is a function satisfying the conditions . For example, if A is the ring of integers, these properties are satisfied with ν(n) the largest value of k such that 2k divides n. ... 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. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... The term global field refers to either of the following: a number field, i. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...

A local field of characteristic 0 is always a finite extension of the field Q_p of p-adic numbers for some prime p. The word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). ... In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ... This article may be too technical for most readers to understand. ...

A local field of characteristic p can always be realized as the field of Laurent series in one variable with coefficients in a finite field (also of characteristic p). In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

Any local field comes equipped with a metric space topology defined by its valuation. Suppose this valuation is denoted v. Since v is a discrete valuation, the set of all values v(x), where x is an element of F, is equal to the integers. The collection of elements of F with non-negative valuation form a compact open subring R of F: x is in R if and only if v(x)≥0. One usually thinks of R as the ring of integers in F. It is a discrete valuation ring with quotient field F. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Valuation can mean: Valuation (finance) Valuation (mathematics) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, a discrete valuation on a commutative ring A is a function satisfying the conditions . For example, if A is the ring of integers, these properties are satisfied with ν(n) the largest value of k such that 2k divides n. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a discrete valuation ring (DVR) is a particular kind of commutative ring that is a local ring, which satisfies conditions that in algebraic geometry come from non-singularity of a point on an algebraic curve. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ...

If F is Q_p, then R is the ring of p-adic integers Z_p; if F is a field of Laurent series, then R is the corresponding ring of power series (those Laurent series without any negative-degree terms). In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

The ring R has only one maximal ideal, which is given by the collection M of elements of F with strictly positive valuation. In the valuation topology of F, M is an open subset. From this it is easy to see that the quotient ring R/M is finite: in general, for any R and M, one can write R as the disjoint union of the distinct residue classes modulo M. But R is a topological ring in this case, so the openness of M implies that all residue classes modulo M are open as well, since they are "shifted versions" of M. So they form an open cover of the compact ring R. By the definition of compactness, some finite subset of the residue classes must cover R, but this means there must only be finitely many residue classes in total, since they are disjoint. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... The word modulo is the Latin ablative of modulus. ... The word modulo is the Latin ablative of modulus. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... Several specialized usages of the terms compact and compactness exist. ...

Finite extensions of local fields are again local fields. If K is a Galois extension of F with Galois group G and valuation ring S, any element g of G sends S to itself. There is a natural metric on G in which the distance between elements g and h is given by the maximum distance between g(s) and h(s) as s ranges over S. This metric gives rise to a filtration on G by normal subgroups called ramification groups. As the quotient of consecutive ramification groups is abelian, Galois groups of local fields are always solvable. The abelian Galois extensions of local fields are of particular interest and form the subject of local classfield theory. In mathematics, Galois theory is a branch of abstract algebra. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with an index set I that is a totally ordered set, subject only to the condition that if i ≤ j in I then Si is contained in Sj. ... In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ... In mathematics, an abelian group is a commutative group, i. ... In mathematics, local classfield theory is the study in number theory of the abelian extensions of local fields. ...

The best single reference for local fields is the book Local Fields by Jean-Pierre Serre, volume GTM 67, published by Springer-Verlag (ISBN 0387904247). Strangely enough, the term 'local field' does not actually appear anywhere in the book. Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ...

See also:

• Ring (algebra)
• Ring ideal
• Topological space
• Compactness
• Modular arithmetic