See subfield in All languages combined, or Wiktionary
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"etymology_text": "From sub- + field.",
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{
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"A smaller, more specialized area of study or occupation within a larger one"
],
"id": "en-subfield-en-noun-rOoOM2i2",
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{
"_dis1": "93 7",
"code": "de",
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"sense": "specialized area of study",
"tags": [
"neuter"
],
"word": "Teilgebiet"
},
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{
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},
{
"_dis1": "93 7",
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"sense": "specialized area of study",
"word": "ਉੱਪਖੇਤਰ"
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]
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{
"ref": "1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.1, page 394:",
"text": "Let us describe in general the subfield generated by a given element. Let K be a given field, F a subfield of K, and c an element of K. Consider those elements of K which are given by polynomial expressions of the form\n(1)#92;qquadf(c)#61;a#95;0#43;a#95;1c#43;a#95;2c²#43;...#43;a#95;ncⁿ#92;qquad#92;qquad#92;mbox#123;(each#125;a#95;i#92;mbox#123;in#125;F#92;mbox#123;).#125;\n[...]\n If f(c) and g(c) ≠ 0 are polynomial expressions like (1), their quotient f(c)/g(c) is an element of K, called a rational expression in c with coefficients in F. The set of all such quotients is a subfield; it is the field generated by F and c and is conventionally denoted by F(c), with round brackets.",
"type": "quote"
},
{
"ref": "1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.2, page 397:",
"text": "We are now in a position to describe the subfield of K generated by F and our algebraic element u. This subfield F(u) clearly contains the subdomain F[u] of all elements expressible as polynomials f(u) with coefficients in F (cf. (1)). Actually, this domain F[u] is a subfield of K. Indeed, let us find an inverse for any element f(u) ≠ 0 in F[u]. [...] This shows that F[u] is a subfield of K.\n Since, conversely, every subfield of K which contains F and u evidently contains every polynomial f(u) in F[u], we see that F[u] is the subfield of K generated by F and u.",
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"A subring of a field, containing the multiplicative identity and closed under inversion."
],
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"(algebra) A subring of a field, containing the multiplicative identity and closed under inversion."
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{
"_dis1": "6 94",
"code": "pl",
"lang": "Polish",
"sense": "algebraic structure",
"tags": [
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"word": "podciało"
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"ref": "1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.1, page 394:",
"text": "Let us describe in general the subfield generated by a given element. Let K be a given field, F a subfield of K, and c an element of K. Consider those elements of K which are given by polynomial expressions of the form\n(1)#92;qquadf(c)#61;a#95;0#43;a#95;1c#43;a#95;2c²#43;...#43;a#95;ncⁿ#92;qquad#92;qquad#92;mbox#123;(each#125;a#95;i#92;mbox#123;in#125;F#92;mbox#123;).#125;\n[...]\n If f(c) and g(c) ≠ 0 are polynomial expressions like (1), their quotient f(c)/g(c) is an element of K, called a rational expression in c with coefficients in F. The set of all such quotients is a subfield; it is the field generated by F and c and is conventionally denoted by F(c), with round brackets.",
"type": "quote"
},
{
"ref": "1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.2, page 397:",
"text": "We are now in a position to describe the subfield of K generated by F and our algebraic element u. This subfield F(u) clearly contains the subdomain F[u] of all elements expressible as polynomials f(u) with coefficients in F (cf. (1)). Actually, this domain F[u] is a subfield of K. Indeed, let us find an inverse for any element f(u) ≠ 0 in F[u]. [...] This shows that F[u] is a subfield of K.\n Since, conversely, every subfield of K which contains F and u evidently contains every polynomial f(u) in F[u], we see that F[u] is the subfield of K generated by F and u.",
"type": "quote"
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"A subring of a field, containing the multiplicative identity and closed under inversion."
],
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"code": "de",
"lang": "German",
"sense": "specialized area of study",
"tags": [
"neuter"
],
"word": "Teilgebiet"
},
{
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"sense": "specialized area of study",
"tags": [
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"word": "poddziedzina"
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"sense": "specialized area of study",
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"word": "subárea"
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"word": "subcampo"
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"roman": "uppakhetar",
"sense": "specialized area of study",
"word": "ਉੱਪਖੇਤਰ"
},
{
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"word": "subfield"
}
Download raw JSONL data for subfield meaning in English (4.1kB)
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2025-03-30 from the enwiktionary dump dated 2025-03-21 using wiktextract (fef8596 and 633533e). 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.
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