# 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. 1.

Every field has the trivial valuation.

2. 2.

The trivial valuation is non-archimedean.

3. 3.

The valuation ring of the trivial valuation is the whole field and the corresponding maximal ideal is the zero ideal.

4. 4.

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

5. 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  $|1|\neq 1$.  Consequently,  $|x|=|a^{m}|=|a|^{m}=1^{m}=1$  for all non-zero elements $x$.)

6. 6.

Every algebraic extension of finite fields has only the trivial valuation, but every field of characteristic 0 has non-trivial valuations.

Title trivial valuation TrivialValuation 2013-03-22 14:20:23 2013-03-22 14:20:23 pahio (2872) pahio (2872) 16 pahio (2872) Definition msc 12J20 msc 11R99 IndependenceOfTheValuations KrullValuation