PlanetMath: exponent valuation

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

exponent valuation

(Definition)

Definition. A function $\nu$ defined in a field is called an exponent valuation or shortly an exponent of the field, if it satisfies the following conditions:

$\nu(0) = \infty$ and $\nu(\alpha)$ runs all rational integers when $\alpha$ runs the non-zero elements of .
$\nu(\alpha\beta) = \nu(\alpha)+\nu(\beta)$ .
$\nu(\alpha+\beta) \geqq \min\{\nu(\alpha),\,\nu(\beta)\}$ .

Note that because of the discrete value set $\mathbb{Z}$ , an exponent valuation belongs to the discrete valuations, and because of notational causes, to the order valuations.

Properties.
$\nu(1) = 0$
$\nu(-\alpha) = \nu(\alpha)$
$\displaystyle\nu\left(\frac{\alpha}{\beta}\right) = \nu(\alpha)\!-\!\nu(\beta)$
$\nu(\alpha^n) = n\,\nu(\alpha)$
$\nu(\alpha_1+\ldots+\alpha_n) \geqq \min\{\nu(\alpha),\,\ldots,\,\nu(\alpha_n)\}$
$\nu(\alpha+\beta) = \min\{\nu(\alpha),\,\nu(\beta)\}$ if $\;\;\;\nu(\alpha) \neq \nu(\beta)$

Example. If an integral domain $\mathcal{O}$ has a divisor theory $\mathcal{O}^* \to \mathfrak{D}$ , then for each prime divisor $\mathfrak{p}$ there is an exponent valuation $\nu_{\mathfrak{p}}$ of the quotient field of $\mathcal{O}$ . It is given by

$\begin{displaymath} \nu_{\mathfrak{p}}(\alpha)\, := \, \begin{cases} & \infty \q... ...\mid (\alpha)\} \;\; \mbox{ when } \,\alpha \neq 0; \end{cases}\end{displaymath}$

$\displaystyle \nu_{\mathfrak{p}}(\xi) \,:=\, \nu_{\mathfrak{p}}(\alpha)-\nu_{\mathfrak{p}}(\beta) \;$ when $\displaystyle \; \xi = \frac{\alpha}{\beta}$ with $\displaystyle \,\alpha,\,\beta \in \mathcal{O}^*.$

Hence, $\mathfrak{p}^{\nu_{\mathfrak{p}}(\alpha)}$ strictly divides $\alpha$ . Apparently, $\nu_{\mathfrak{p}}(\xi)$ does not depend on the quotient form $\frac{\alpha}{\beta}$ for $\xi$ . It is not hard to show that $\nu_{\mathfrak{p}}$ defined above is an exponent of the field

Different prime divisors $\mathfrak{p}$ and $\mathfrak{q}$ determine different exponents $\nu_{\mathfrak{p}}$ and $\nu_{\mathfrak{q}}$ , since the condition 3 of the definition of divisor theory guarantees such an element $\gamma$ of $\mathcal{O}$ which in divisible by $\mathfrak{p}$ but not by $\mathfrak{q}$ ; then $\nu_{\mathfrak{p}}(\gamma) \geqq 1$ , $\nu_{\mathfrak{q}}(\gamma) = 0$ .

Theorem. Let $\nu_1,\,\ldots,\,\nu_r$ be different exponents of a field . Then for arbitrary set $n_1,\,\ldots,\,n_r$ of integers, there exists in an element $\xi$ such that

$\displaystyle \nu_1(\xi) = n_1,\;\;\ldots,\;\;\nu_r(\xi) = n_r.$

The proof of this theorem is found in [1].

Bibliography

1: S. BOREWICZ & I. SAFAREVIC: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).

"exponent valuation" is owned by pahio.

(view preamble)

Other names:

exponent of field

Also defines:

exponent of a field, exponent of the field

Keywords:

exponent

This object's parent.

Attachments:: ring of exponent (Definition) by pahio
continuation of exponent (Definition) by pahio

Log in to rate this entry.
(view current ratings)

Cross-references: divisible, quotient, strictly divides, quotient field, prime divisor, divisor theory, integral domain, properties, order valuations, discrete valuations, discrete, integers, rational, exponent, field, function
There are 9 references to this entry.

This is version 8 of exponent valuation, born on 2008-04-15, modified 2008-05-16.
Object id is 10503, canonical name is ExponentValuation2.
Accessed 477 times total.

Classification:

AMS MSC:	11R99 (Number theory :: Algebraic number theory: global fields :: Miscellaneous)
	12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
	13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations)
	13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact