discrete valuation

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discrete valuation

Wikipedia

Noun

discrete valuation (plural discrete valuations)

1. (algebra, field theory) A valuation (on some field) that takes integer values (including infinity).
   • 2000, Freddy Van Oystaeyen, Algebraic Geometry for Associative Algebras, CRC Press, page 173:

     Therefore the theory branches in two essential parts : first the study of discrete valuations of the Weyl field ${\displaystyle D_{1}(K)}$ extending a discrete valuation of ${\displaystyle K}$, secondly the discrete valuations of ${\displaystyle D_{1}(K)}$ having the purely transcendental field ${\displaystyle K(t)}$ as a residue field.

   • 2004, Farahat S. Aly, Freddy Van Oystaeyen, “Hopf Order Module Algebra Orders”, in Stefaan Caenepeel, Fred Van Oystaeyen, editors, Hopf Algebras in Noncommutative Geometry and Physics, CRC Press, page 21:

     One of the few possibilities to extend some arithmetic aspects of number theory to more abstract fields is via valuation theory and, as a first step, we may restrict attention to discrete valuations.

   • 2011, Ricardo Portilla, “Parametrizing nilpotent orbits in p-adic symmetric spaces”, in Robert S. Doran, Paul J. Sally, Jr., Loren Spice, editors, Harmonic Analysis on Reductive, p-adic Groups, American Mathematical Society, page 201:

     Let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field.

Derived terms

• discrete valuation field
• discrete valuation ring

See also

• p-adic absolute value
• p-adic norm
• residue field
• valuation algebra
• valuation domain
• valuation ring

Further reading

• Valuation (algebra) on Wikipedia.Wikipedia
• Information algebra on Wikipedia.Wikipedia
• Valuation on Wolfram MathWorld
• Valuation on Encyclopedia of Mathematics
• Discretely-normed ring on Encyclopedia of Mathematics