"discrete valuation" meaning in All languages combined

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  "lang_code": "en",
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      "derived": [
        {
          "word": "discrete valuation field"
        },
        {
          "word": "discrete valuation ring"
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      "examples": [
        {
          "ref": "2000, Freddy Van Oystaeyen, Algebraic Geometry for Associative Algebras, CRC Press, page 173",
          "text": "Therefore the theory branches in two essential parts : first the study of discrete valuations of the Weyl field D#x5F;1(K) extending a discrete valuation of K, secondly the discrete valuations of D#x5F;1(K) having the purely transcendental field K(t) as a residue field.",
          "type": "quotation"
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        {
          "ref": "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",
          "text": "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.",
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          "ref": "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",
          "text": "Let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field.",
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        "A valuation (on some field) that takes integer values (including infinity)."
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        "(algebra, field theory) A valuation (on some field) that takes integer values (including infinity)."
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          "word": "p-adic absolute value"
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          "word": "p-adic norm"
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        {
          "word": "valuation algebra"
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        {
          "word": "valuation domain"
        },
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          "word": "valuation ring"
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        "discrete valuation"
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  "word": "discrete valuation"
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          "text": "Therefore the theory branches in two essential parts : first the study of discrete valuations of the Weyl field D#x5F;1(K) extending a discrete valuation of K, secondly the discrete valuations of D#x5F;1(K) having the purely transcendental field K(t) as a residue field.",
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          "ref": "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",
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          "ref": "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",
          "text": "Let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field.",
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