See p-adic absolute value in All languages combined, or Wiktionary
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"text": "According to Ostrowski's theorem, only three kinds of norms are possible for the set of real numbers: the trivial absolute value, the real absolute value, and the p-adic absolute value.ᵂᴾ",
"type": "example"
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"ref": "1993, Seth Warner, Topological Rings, Elsevier (North-Holland), page 8",
"text": "If c#x3E;1, then x#x5C;rightarrowc#x7B;-v#x5F;p(x)#x7D; (with the convention c#x7B;-#x5C;infty#x7D;#x3D;0) is a nonarchimedean absolute value, denoted #x5C;vert..#x5C;vert#x5F;#x7B;p,c#x7D; and called the p-adic absolute value to base c. If c#x3E;1 and d#x3E;1 and if r#x3D;#x5C;log#x5F;cd, then #x5C;vertx#x5C;vert#x5F;#x7B;p,d#x7D;#x3D;#x5C;vertx#x5C;vertʳ#x5F;#x7B;p,c#x7D; for every x#x5C;inK. The p-adic topology on K is the topology defined by the p-adic absolute values.",
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"ref": "1999, Jan-Hendrik Evertse, Hans Peter Schlickewei, “The Absolute Subspace Theorem and linear equations with unknowns from a multiplicative group”, in Kálmán Györy, Henryk Iwaniec, Jerzy Urbanowicz, editors, Number Theory in Progress, Walter de Gruyter, page 121",
"text": "They both gave essentially the same proof, based on the Subspace Theorem (more precisely, Schlickewei's generalisation to p-adic absolute values and number fields [30] of the Subspace Theorem proved by Schmidt in 1972 [41]).",
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"ref": "2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 320",
"text": "It can also be defined as the subset with #x5C;vertx#x5C;vert#x5F;p#x3C;p because the p-adic absolute value takes no values between 1 and p, and therefore #x5C;Z#x5F;p is open.",
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"A norm for the rational numbers, with some prime number p as parameter, such that any rational number of the form pᵏ(a/b) — where a, b and k are integers and a, b and p are coprime — is mapped to the rational number p⁻ᵏ and 0 is mapped to 0. (Note: any nonzero rational number can be reduced to such a form.)"
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"(number theory, field theory) A norm for the rational numbers, with some prime number p as parameter, such that any rational number of the form pᵏ(a/b) — where a, b and k are integers and a, b and p are coprime — is mapped to the rational number p⁻ᵏ and 0 is mapped to 0. (Note: any nonzero rational number can be reduced to such a form.)"
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"text": "According to Ostrowski's theorem, only three kinds of norms are possible for the set of real numbers: the trivial absolute value, the real absolute value, and the p-adic absolute value.ᵂᴾ",
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"text": "If c#x3E;1, then x#x5C;rightarrowc#x7B;-v#x5F;p(x)#x7D; (with the convention c#x7B;-#x5C;infty#x7D;#x3D;0) is a nonarchimedean absolute value, denoted #x5C;vert..#x5C;vert#x5F;#x7B;p,c#x7D; and called the p-adic absolute value to base c. If c#x3E;1 and d#x3E;1 and if r#x3D;#x5C;log#x5F;cd, then #x5C;vertx#x5C;vert#x5F;#x7B;p,d#x7D;#x3D;#x5C;vertx#x5C;vertʳ#x5F;#x7B;p,c#x7D; for every x#x5C;inK. The p-adic topology on K is the topology defined by the p-adic absolute values.",
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"text": "It can also be defined as the subset with #x5C;vertx#x5C;vert#x5F;p#x3C;p because the p-adic absolute value takes no values between 1 and p, and therefore #x5C;Z#x5F;p is open.",
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