# Discrete norm

From Encyclopedia of Mathematics

A norm on a skew-field the group of values of which is isomorphic to the group of integers $ \mathbf Z $.
In such a case the ring is a discretely-normed ring. A discrete norm, more exactly, a discrete norm of height (or rank) $ r $
is also sometimes understood as the norm having as group of values the $ r $-
th direct power of the group $ \mathbf Z $
with the lexicographical order.

#### Comments

This notion is more commonly called a discrete valuation. A discretely-normed ring is usually called a discrete valuation domain. See also Norm on a field; Valuation.

#### References

[a1] | O. Endler, "Valuation theory" , Springer (1972) |

**How to Cite This Entry:**

Discrete norm.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Discrete_norm&oldid=46734

This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article