discrete valuation

English

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 $D_{1}(K)$ extending a discrete valuation of $K$, secondly the discrete valuations of $D_{1}(K)$ having the purely transcendental field $K(t)$ as a residue field.

   • 2004, Farahat S. Aly, Freddy Van Oystaeyen, Hopf Order Module Algebra Orders, 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, 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