PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (190)
Orphanage
Unclass'd (1)
Unproven (496)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
discrete valuation (Definition)

A discrete valuation on a field $ K$ is a valuation $ \vert\cdot\vert: K \to \mathbb{R}$ whose image is a discrete subset of $ \mathbb{R}$.

For any field $ K$ with a discrete valuation $ \vert\cdot\vert$, the set

$\displaystyle R := \{x \in K : \vert x\vert \leq 1\} $
is a subring of $ K$ with sole maximal ideal
$\displaystyle M := \{x \in K : \vert x\vert < 1\}, $
and hence $ R$ is a discrete valuation ring. Conversely, given any discrete valuation ring $ R$, the field of fractions $ K$ of $ R$ admits a discrete valuation sending each element $ x \in R$ to $ c^n$, where $ 0 < c < 1$ is some arbitrary fixed constant and $ n$ is the order of $ x$, and extending multiplicatively to $ K$.

Note: Discrete valuations are often written additively instead of multiplicatively; under this alternate viewpoint, the element $ x$ maps to $ \log_c\vert x\vert$ (in the above notation) instead of just $ \vert x\vert$. This transformation reverses the order of the absolute values (since $ c < 1$), and sends the element $ 0 \in K$ to $ \infty$. It has the advantage that every valuation can be normalized by a suitable scalar multiple to take values in the integers.



"discrete valuation" is owned by djao.
(view preamble)

View style:

See Also: discrete valuation ring, valuation

Other names:  rank one valuations, rank-one valuations
Log in to rate this entry.
(view current ratings)

Cross-references: integers, scalar multiple, absolute values, transformation, maps, order, fixed, field of fractions, discrete valuation ring, maximal ideal, subring, discrete subset, image, valuation, field
There are 6 references to this entry.

This is version 3 of discrete valuation, born on 2003-10-06, modified 2005-07-24.
Object id is 4761, canonical name is DiscreteValuation.
Accessed 3802 times total.

Classification:
AMS MSC: 13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)
 12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)