trivial valuation
The trivial valuation of a field is the Krull valuation of such that and for other elements of .
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 be the primitive element of the multiplicative group of the field, which is cyclic (http://planetmath.org/CyclicGroup). If is any valuation of the field, then one must have since otherwise . Consequently, for all non-zero elements .)
-
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 |
---|---|
Canonical name | TrivialValuation |
Date of creation | 2013-03-22 14:20:23 |
Last modified on | 2013-03-22 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 |