trivial valuationTrivialValuation2004-04-29 03:00:332006-12-18 12:55:16Definitionvaluation% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe {\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 $K$.
\subsection*{Properties}
\begin{enumerate}
\item Every field has the trivial valuation.
\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 field is \PMlinkname{complete}{Complete} with respect to (the metric given by) its trivial valuation.
\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 $x$.)
\item Every algebraic extension of finite fields has only the trivial valuation, but every field of characteristic 0 has non-trivial valuations.
\end{enumerate}