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ring of exponent

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\begin{document}
\PMlinkescapeword{exponents} \PMlinkescapeword{exponent}

\textbf{Definition.}\, 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 \PMlinkescapetext{{\em ring of the exponent}} $\nu$.\, It is, naturally, an integral domain.\, Its elements are called \PMlinkescapetext{{\em integral with respect to}} $\nu$.\\

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

\textbf{Theorem 2.}\, The ring $\mathcal{O}_\nu$ \PMlinkescapetext{contains} 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 \PMlinkescapetext{fixed} $\pi$ 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.\\

\textbf{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 \,$\nu(\varepsilon) = 0$.

\textbf{Remark 2.}\, In an algebraically closed field $\Omega$, there are no \PMlinkname{exponents}{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 \PMlinkname{root}{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.\\

\textbf{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 \PMlinkname{unique factorisation}{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 \PMlinkescapetext{fixed} coprime prime elements of $\mathfrak{O}$ satisfying
\[
\nu_i(\pi_j) \;=\; \delta_{ij} \;=\;  
\begin{cases}
& 1 \;\;\mbox{for  }\, i = j,\\
& 0 \;\;\mbox{for  }\, i \neq j.
\end{cases} 
\]

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