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Revision difference : trivial valuation |
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Version 12 |
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The {\em trivial valuation} of a field $K$ is the Krull valuation\, $|\cdot|$\, of $K$ such that\, $|0| = 0$\, and\, $|x| = 1$\, for other elements $x$ of $K$.
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The {\em trivial valuation} of a field $K$ is the Krull valuation \,$|\cdot|$\, of $K$ such that \,$|0| = 0$\, and \,$|x| = 1$\, for other elements $x$ of $K$.
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| \subsection*{Properties} |
\subsection*{Properties} |
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| \begin{enumerate} |
\begin{enumerate} |
| \item Every field has the trivial valuation. |
\item Every field has the trivial valuation. |
| \item The trivial valuation is non-archimedean. |
\item The trivial valuation is non-archimedean. |
| \item The valuation ring of the trivial valuation is the whole field and the corresponding maximal ideal is the zero ideal. |
\item The valuation ring of the trivial valuation is the whole field and the corresponding maximal ideal is the zero ideal. |
| \item The field is \PMlinkname{complete}{Complete} with respect to (the metric given by) its trivial valuation. |
\item The field is \PMlinkname{complete}{Complete} with respect to (the metric given by) its trivial valuation. |
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\item A finite field has only the trivial valuation.\, (Let $a$ be the primitive element of the multiplicative group of the field, which is \PMlinkname{cyclic}{CyclicGroup}.\, If \,$|\cdot|$\, is any valuation of the field, then one must have\, $|a| = 1$\, since otherwise\, $|1| \neq 1$.\, Consequently,\, $|x| = |a^m| = |a|^m = 1^m = 1$\, for all non-zero elements $x$.)
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\item A finite field has only the trivial valuation. (Let $a$ be the primitive element of the multiplicative group of the field, which is \PMlinkname{cyclic}{CyclicGroup}. \,If \,$|\cdot|$\, is any valuation of the field, then one must have \,$|a| = 1$\, since otherwise $|1| \neq 1$. \,Consequently, \,$|x| = |a^m| = |a|^m = 1^m = 1$\, for all non-zero elements $x$.)
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\item Every algebraic extension of finite fields has only the trivial valuation, but every field of characteristic 0 has non-trivial valuations.
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\item Every infinite field has at least one non-trivial valuation.
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| \end{enumerate} |
\end{enumerate} |
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