A special case of this are the discrete valuation rings mentioned earlier.

K is complete under a discrete valuation with finite residue field.

Then Z is the discrete valuation ring corresponding to ν.

Z is an example of a complete discrete valuation ring of mixed characteristic.

This is also a discrete valuation ring.

The discrete valuation case is much like the complex unit disk, for these purposes.)

For example, the spectrum of a discrete valuation ring consists of two points and is connected.

Dedekind domains that are not fields (for example, discrete valuation rings) have dimension one.

In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field.

Any discrete valuation ring.

There is a similar construction with Z replaced by any complete discrete valuation ring with finite residue class field.

This theory concerns extensions of "local" (i.e., complete for a discrete valuation) fields with finite residue field.

All principal ideal domains and therefore all discrete valuation rings are Dedekind domains.

Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring.

Discrete valuation ring, The Encyclopaedia of Mathematics.

Such topologies are not the topologies associated with discrete valuations of rank , if .

(For a discrete valuation ring the topological space in question is the Sierpinski space of topologists.

This can be done over more general fields by using the set of discrete valuation rings of the field as a substitute for the Riemann surface.

The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.

One dimensional local fields are usually considered in the valuation topology, in which the discrete valuation is used to define open sets.

An important class of local rings are discrete valuation rings, which are local principal ideal domains that are not fields.

Similarly, there is a discrete valuation of rank on almost all -dimensional local fields, associated to a choice of local parameters of the field.

A field with a non-trivial discrete valuation is called a discrete valuation field.

(DD2) is Noetherian, and the localization at each maximal ideal is a Discrete Valuation Ring.