trivial valuation
The 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 $K$.
Properties

1.
Every field has the trivial valuation.

2.
The trivial valuation is nonarchimedean.

3.
The valuation ring^{} of the trivial valuation is the whole field and the corresponding maximal ideal^{} is the zero ideal^{}.

4.
The field is complete^{} (http://planetmath.org/Complete) with respect to (the metric given by) its trivial valuation.

5.
A finite field^{} has only the trivial valuation. (Let $a$ be the primitive element^{} of the multiplicative group^{} of the field, which is cyclic (http://planetmath.org/CyclicGroup). If $\cdot $ is any valuation of the field, then one must have $a=1$ since otherwise $1\ne 1$. Consequently, $x={a}^{m}={a}^{m}={1}^{m}=1$ for all nonzero elements $x$.)

6.
Every algebraic extension^{} of finite fields has only the trivial valuation, but every field of characteristic^{} 0 has nontrivial valuations.
Title  trivial valuation 

Canonical name  TrivialValuation 
Date of creation  20130322 14:20:23 
Last modified on  20130322 14:20:23 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  16 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 12J20 
Classification  msc 11R99 
Related topic  IndependenceOfTheValuations 
Related topic  KrullValuation 