Local field

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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.

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

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.

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).

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.

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.

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References

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