See p-adic absolute value in All languages combined, or Wiktionary
{
"forms": [
{
"form": "p-adic absolute values",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"1": "p-adic absolute values",
"head": "p-adic absolute value"
},
"expansion": "p-adic absolute value (plural p-adic absolute values)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English links with redundant target parameters",
"parents": [
"Links with redundant target parameters",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English links with redundant wikilinks",
"parents": [
"Links with redundant wikilinks",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English terms containing italics",
"parents": [
"Terms containing italics",
"Terms by orthographic property",
"Terms by lexical property"
],
"source": "w"
},
{
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with DISPLAYTITLE conflicts",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Number theory",
"orig": "en:Number theory",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"examples": [
{
"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"
},
{
"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.",
"type": "quote"
},
{
"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]).",
"type": "quote"
},
{
"ref": "2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 320:",
"text": "It ^([#x5C;Z#x5F;p]) can also be defined as the subset ^([of #x5C;Q#x5F;p]) 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.",
"type": "quote"
}
],
"glosses": [
"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.)"
],
"hypernyms": [
{
"word": "norm"
}
],
"id": "en-p-adic_absolute_value-en-noun-i4Z9nAds",
"links": [
[
"number theory",
"number theory"
],
[
"norm",
"norm"
],
[
"rational number",
"rational number"
],
[
"prime number",
"prime number"
],
[
"parameter",
"parameter"
],
[
"integer",
"integer"
],
[
"coprime",
"coprime"
],
[
"map",
"map"
]
],
"qualifier": "field theory",
"raw_glosses": [
"(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.)"
],
"related": [
{
"word": "p-adic order"
},
{
"word": "p-adic valuation"
},
{
"word": "p-adic ultrametric"
}
],
"synonyms": [
{
"word": "p-adic norm"
}
],
"topics": [
"mathematics",
"number-theory",
"sciences"
]
}
],
"word": "p-adic absolute value"
}
{
"forms": [
{
"form": "p-adic absolute values",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"1": "p-adic absolute values",
"head": "p-adic absolute value"
},
"expansion": "p-adic absolute value (plural p-adic absolute values)",
"name": "en-noun"
}
],
"hypernyms": [
{
"word": "norm"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "p-adic order"
},
{
"word": "p-adic valuation"
},
{
"word": "p-adic ultrametric"
}
],
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English links with redundant target parameters",
"English links with redundant wikilinks",
"English multiword terms",
"English nouns",
"English terms containing italics",
"English terms with quotations",
"English terms with usage examples",
"Pages with 1 entry",
"Pages with DISPLAYTITLE conflicts",
"Pages with entries",
"en:Number theory"
],
"examples": [
{
"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"
},
{
"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.",
"type": "quote"
},
{
"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]).",
"type": "quote"
},
{
"ref": "2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 320:",
"text": "It ^([#x5C;Z#x5F;p]) can also be defined as the subset ^([of #x5C;Q#x5F;p]) 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.",
"type": "quote"
}
],
"glosses": [
"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.)"
],
"links": [
[
"number theory",
"number theory"
],
[
"norm",
"norm"
],
[
"rational number",
"rational number"
],
[
"prime number",
"prime number"
],
[
"parameter",
"parameter"
],
[
"integer",
"integer"
],
[
"coprime",
"coprime"
],
[
"map",
"map"
]
],
"qualifier": "field theory",
"raw_glosses": [
"(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.)"
],
"topics": [
"mathematics",
"number-theory",
"sciences"
]
}
],
"synonyms": [
{
"word": "p-adic norm"
}
],
"word": "p-adic absolute value"
}
Download raw JSONL data for p-adic absolute value meaning in English (3.6kB)
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.