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A norm on a skew-field the group of values of which is isomorphic to the group of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033100/d0331001.png" />. In such a case the ring is a [[Discretely-normed ring|discretely-normed ring]]. A discrete norm, more exactly, a discrete norm of height (or rank) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033100/d0331002.png" /> is also sometimes understood as the norm having as group of values the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033100/d0331003.png" />-th direct power of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033100/d0331004.png" /> with the lexicographical order.
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A norm on a skew-field the group of values of which is isomorphic to the group of integers  $  \mathbf Z $.
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In such a case the ring is a [[Discretely-normed ring|discretely-normed ring]]. A discrete norm, more exactly, a discrete norm of height (or rank)  $  r $
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is also sometimes understood as the norm having as group of values the  $  r $-
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th direct power of the group  $  \mathbf Z $
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with the lexicographical order.
  
 
====Comments====
 
====Comments====

Latest revision as of 19:36, 5 June 2020


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=12162
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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