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| Title of object: |
trivial valuation |
| Canonical Name: |
TrivialValuation |
| Type: |
Definition |
| Created on: |
2004-04-29 03:00:33 |
| Modified on: |
2004-04-29 10:17:44 |
| Classification: |
msc:11R99, msc:12J20 |
Preamble:
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Content:
The {\em trivial valuation} of a field $K$ is the mapping $|\cdot|$ from $K$ to $\mathbb{R}$ such that $|0| = 0$ and $|a| = 1$ for other elements $a$ of $K$.
\subsection*{Properties}
\begin{enumerate}
\item The trivial valuation is non-archimedean.
\item Every field has the trivial valuation. (If $a^m = 1$ in a finite field and $|\cdot|$ is its valuation, then $|a|^m = |a^m| = |1| = 1$ and thus $|a| = 1$.)
\item A finite field has only the trivial valuation.
\item Every infinite field has at least one non-trivial valuation.
\end{enumerate} |
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