ring of exponent

Definition.  Let $\nu$ be an exponent valuation of the field $K$.  The subring

$${𝒪}_{\nu }:=\{\alpha \in K\mathrm{⋮}\nu (\alpha )\geqq 0\}$$

of $K$ is called the $\nu$.  It is, naturally, an integral domain.  Its elements are called $\nu$.

Theorem 1.  The ring of the exponent $\nu$ of the field $K$ is integrally closed in $K$.

Theorem 2.  The ring ${𝒪}_{\nu }$ only one prime element $\pi$, when one does not regard associated elements as different.  Any non-zero element $\alpha$ can be represented uniquely with a $\pi$ in the form

$$\alpha =\epsilon {\pi }^m,$$

where $\epsilon$ is a unit of ${𝒪}_{\nu }$ and  $m=\nu (\alpha )\geqq 0$.  This means that $𝒪$ is a UFD.

Remark 1.  The prime elements $\pi$ of the ring ${𝒪}_{\nu }$ are characterised by the equation  $\nu (\pi )=1$  and the units  $\epsilon$ the equation  $\nu (\epsilon )=0$.

Remark 2.  In an algebraically closed field $\mathrm{\Omega }$, there are no exponents (http://planetmath.org/ExponentValuation).  In fact, if there were an exponent $\nu$ of $\mathrm{\Omega }$ and if $\pi$ were a prime element of the ring of the exponent, then, since the equation  $x^2-\pi =0$  would have a root (http://planetmath.org/Equation) $\varrho$ in $\mathrm{\Omega }$, we would obtain  $2\nu (\varrho )=\nu ({\varrho }^2)=\nu (\pi )=1$;  this is however impossible, because an exponent attains only integer values.

Theorem 3.  Let  ${𝔒}_1,\mathrm{\dots },{𝔒}_r$ be the rings of the different exponent valuations ${\nu }_1,\mathrm{\dots },{\nu }_r$ of the field $K$.  Then also the intersection

$$𝔒:=\bigcap _{i=1}^r{𝔒}_i$$

is a subring of $K$ with unique factorisation (http://planetmath.org/UFD).  To be precise, any non-zero element $\alpha$ of $𝔒$ may be uniquely represented in the form

$$\alpha =\epsilon {\pi }_1^{n_1}\mathrm{\cdots }{\pi }_r^{n_r},$$

in which $\epsilon$ is a unit of $𝔒$,  the integers $n_1,\mathrm{\dots },n_r$ are nonnegative and ${\pi }_1,\mathrm{\dots },{\pi }_r$ are coprime prime elements of $𝔒$ satisfying

Title	ring of exponent
Canonical name	RingOfExponent
Date of creation	2013-03-22 17:59:43
Last modified on	2013-03-22 17:59:43
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Definition
Classification	msc 13F30
Classification	msc 13A18
Classification	msc 12J20
Classification	msc 11R99
Related topic	DiscreteValuationRing
Related topic	ValuationRingOfAField
Related topic	LocalRing
Defines	ring of an exponent
Defines	ring of the exponent
Defines	integral with respect to an exponent