Definition. The mapping $|.|\!:\, K\to G$ where $K$ is a field and $G$ an ordered group equipped with zero, is a Krull valuation of $K$ if it has the properties

1. $|x| = 0 \,\,\Leftrightarrow\,\, x = 0$
2. $|xy| = |x|\cdot|y|$
3. $|x+y| \leqq \max\{|x|,\,|y|\}$

Thus the Krull valuation is more general than the usual valuation, which is also characterized as valuation of rank 1 and which has real values. The image $|K\!\smallsetminus\!\{0\}|$ , is called the value group of the Krull valuation; it is abelian. In general, the rank of Krull valuation means the rank of the value group.

We may say that a Krull valuation is non-archimedean.

Some values

• $|1| = 1$ , because the Krull valuation is a group homomorphism from the multiplicative group of $K$ to the ordered group.
• $|-1| = 1$ , because $1 = |(-1)^2| = |-1|^2$ , and 1 is the only element of the ordered group being its own inverse ($S\cap S^{-1} = \varnothing$ .
• $|-x| = |(-1)x| = |-1|\cdot|x| = |x|$

Bibliography

1

EMIL ARTIN: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).