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Local field

In mathematics, a local field is a special type of algebraic field which has the additional property that it is a complete metric space with respect to a discrete valuation[?]. Local fields are important in diverse areas of mathematics, but primarily so in number theory, where they have been used in the application of the local-global principle[?], developed by Hasse[?].

A local field of characteristic 0 is either a finite extension of a field of p-adic numbers or a finite extension of a field of formal Laurent series in one variable with coefficients from a global field[?].

A local field of characteristic p is a finite extension of a field of Laurent series in one variable with coefficients from 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. There are several very nice topological properties of F in this valuation topology, beginning with the existence of a maximal compact subring, which will be described next.

This maximal compact subring R of F can be defined as the collection of elements of F with nonpositive valuation: x is in R if and only if v(x) is less than or equal to zero. R is sometimes called the valuation ring of F. This ring is obviously an integral domain (because it is contained in a field, hence has no zero divisors), and its quotient field is F.

If F is the p-adic numbers, then R consists of the p-adic integers; 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).

R is a very special kind of ring. It has only one maximal ideal, which is given by the collection M of elements of F with strictly negative 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.

What is remarkable about all this is that it is very unlike the behavior we are used to from the real numbers, in that there is no compact subring of the reals.

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