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 non-archimedean.

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 non-zero elements $x$.)

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