# ring of exponent

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

 $\mathcal{O}_{\nu}\;:=\;\{\alpha\in K\,\vdots\;\;\nu(\alpha)\geqq 0\}$

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 $\mathcal{O}_{\nu}$ 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

 $\alpha\;=\;\varepsilon\pi^{m},$

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

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

Remark 2.  In an algebraically closed field $\Omega$, there are no exponents (http://planetmath.org/ExponentValuation).  In fact, if there were an exponent $\nu$ of $\Omega$ 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 $\Omega$, 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  $\mathfrak{O}_{1},\,\ldots,\,\mathfrak{O}_{r}$ be the rings of the different exponent valuations $\nu_{1},\,\ldots,\,\nu_{r}$ of the field $K$.  Then also the intersection

 $\mathfrak{O}\;:=\;\bigcap_{i=1}^{r}\mathfrak{O}_{i}$

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

 $\alpha\;=\;\varepsilon\pi_{1}^{n_{1}}\cdots\pi_{r}^{n_{r}},$

in which $\varepsilon$ is a unit of $\mathfrak{O}$,  the integers $n_{1},\,\ldots,\,n_{r}$ are nonnegative and $\pi_{1},\,\ldots,\,\pi_{r}$ are coprime prime elements of $\mathfrak{O}$ satisfying

 $\nu_{i}(\pi_{j})\;=\;\delta_{ij}\;=\;\begin{cases}&1\;\;\mbox{for }\,i=j,\\ &0\;\;\mbox{for }\,i\neq j.\end{cases}$
 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