Local field

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 R of real numbers and the field C 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.

Related Topics:
Field - Complete metric space - Discrete valuation - Locally compact - Real number - Complex number - Number theory - Global field - Number field

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A local field of characteristic 0 is always a finite extension of the field Qp of p-adic numbers for some prime p.

Related Topics:
Characteristic - Finite extension - P-adic number

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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).

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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.

Related Topics:
Metric space - Topology - Valuation - Discrete valuation - Integer - Discrete valuation ring - Quotient field

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If F is Qp, then R is the ring of p-adic integers Zp; 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).

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The ring R has only one prime ideal, which is given by the collection M of elements of F with strictly positive valuation, making R a local ring. 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.

Related Topics:
Prime ideal - Local ring - Open subset - Quotient ring - Disjoint - Residue class - Modulo - Open cover - Compactness

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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.

Related Topics:
Galois extension - Galois group - Filtration - Ramification group - Abelian - Local classfield theory

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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.

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See also:

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