See Galois field in All languages combined, or Wiktionary
Download JSON data for Galois field meaning in English (4.2kB)
{
"etymology_templates": [
{
"args": {
"1": "mathematician",
"2": "",
"3": "",
"4": "",
"5": ""
},
"expansion": "mathematician",
"name": "named-after/list"
},
{
"args": {},
"expansion": "|",
"name": "!"
},
{
"args": {
"1": "en",
"2": "Évariste Galois"
},
"expansion": "Évariste Galois",
"name": "lang"
},
{
"args": {
"1": "en",
"2": "Évariste Galois",
"born": "1811",
"died": "1832",
"nat": "French",
"occ": "mathematician",
"wplink": "="
},
"expansion": "Named after French mathematician Évariste Galois (1811–1832)",
"name": "named-after"
}
],
"etymology_text": "Named after French mathematician Évariste Galois (1811–1832).",
"forms": [
{
"form": "Galois fields",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "Galois field (plural Galois fields)",
"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 entries with language name categories using raw markup",
"parents": [
"Entries with language name categories using raw markup",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English terms with non-redundant non-automated sortkeys",
"parents": [
"Terms with non-redundant non-automated sortkeys",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"examples": [
{
"text": "The Galois field #x5C;mathrm#x7B;GF#x7D;(pⁿ) has order pⁿ and characteristic p.",
"type": "example"
},
{
"text": "The Galois field #x5C;mathrm#x7B;GF#x7D;(pⁿ) is a finite extension of the Galois field #x5C;mathrm#x7B;GF#x7D;(p) and the degree of the extension is n.",
"type": "example"
},
{
"text": "The multiplicative subgroup of a Galois field is cyclic.",
"type": "example"
},
{
"text": "A Galois field #x5C;mathbb#x7B;F#x7D;#x5F;#x7B;pⁿ#x7D; is isomorphic to the quotient of the polynomial ring #x5C;mathbb#x7B;F#x7D;#x5F;p adjoin x over the ideal generated by a monic irreducible polynomial of degree n. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: #x5C;mathbb#x7B;F#x7D;#x5F;#x7B;pⁿ#x7D;#x5C;cong#x7B;#x5C;mathbb#x7B;F#x7D;#x5F;p#x5B;x#x5D;#x5C;over(#x5C;hatf#x5F;n(x))#x7D;.",
"type": "example"
},
{
"text": "1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page,\nA field with a finite number of elements is called a Galois field.\nThe number of elements of the prime field k contained in a Galois field K is finite, and is therefore a natural prime p."
},
{
"ref": "2001, Joseph E. Bonin, A Brief Introduction To Matroid Theory, retrieved 2016-05-05",
"text": "The case of most interest to us will be that in which F is a finite field, the Galois field GF(q) for some prime power q. If q is prime, this field is #x5C;mathbb#x7B;Z#x7D;#x5F;q, the integers 0,1,#x5C;dots,q-1 with arithmetic modulo q.",
"type": "quotation"
},
{
"text": "2006, Debojyoti Battacharya, Debdeep Mukhopadhyay, D. RoyChowdhury, A Cellular Automata Based Approach for Generation of Large Primitive Polynomial and Its Application to RS-Coded MPSK Modulation, Samira El Yacoubi, Bastien Chopard, Stefania Bandini (editors), Cellular Automata: 7th International Conference, Proceedings, Springer, LNCS 4173, page 204,\nGeneration of large primitive polynomial over a Galois field has been a topic of intense research over the years. The problem of finding a primitive polynomial over a Galois field of a large degree is computationaly expensive and there is no deterministic algorithm for the same."
}
],
"glosses": [
"A finite field; a field that contains a finite number of elements."
],
"hypernyms": [
{
"word": "perfect field"
}
],
"id": "en-Galois_field-en-noun-XVm7itKZ",
"links": [
[
"algebra",
"algebra"
],
[
"finite field",
"finite field"
],
[
"finite",
"finite"
],
[
"elements",
"elements"
]
],
"raw_glosses": [
"(algebra) A finite field; a field that contains a finite number of elements."
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"wikipedia": [
"Finite field"
]
}
],
"word": "Galois field"
}
{
"etymology_templates": [
{
"args": {
"1": "mathematician",
"2": "",
"3": "",
"4": "",
"5": ""
},
"expansion": "mathematician",
"name": "named-after/list"
},
{
"args": {},
"expansion": "|",
"name": "!"
},
{
"args": {
"1": "en",
"2": "Évariste Galois"
},
"expansion": "Évariste Galois",
"name": "lang"
},
{
"args": {
"1": "en",
"2": "Évariste Galois",
"born": "1811",
"died": "1832",
"nat": "French",
"occ": "mathematician",
"wplink": "="
},
"expansion": "Named after French mathematician Évariste Galois (1811–1832)",
"name": "named-after"
}
],
"etymology_text": "Named after French mathematician Évariste Galois (1811–1832).",
"forms": [
{
"form": "Galois fields",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "Galois field (plural Galois fields)",
"name": "en-noun"
}
],
"hypernyms": [
{
"word": "perfect field"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English entries with language name categories using raw markup",
"English eponyms",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with non-redundant non-automated sortkeys",
"English terms with quotations",
"English terms with usage examples",
"en:Algebra"
],
"examples": [
{
"text": "The Galois field #x5C;mathrm#x7B;GF#x7D;(pⁿ) has order pⁿ and characteristic p.",
"type": "example"
},
{
"text": "The Galois field #x5C;mathrm#x7B;GF#x7D;(pⁿ) is a finite extension of the Galois field #x5C;mathrm#x7B;GF#x7D;(p) and the degree of the extension is n.",
"type": "example"
},
{
"text": "The multiplicative subgroup of a Galois field is cyclic.",
"type": "example"
},
{
"text": "A Galois field #x5C;mathbb#x7B;F#x7D;#x5F;#x7B;pⁿ#x7D; is isomorphic to the quotient of the polynomial ring #x5C;mathbb#x7B;F#x7D;#x5F;p adjoin x over the ideal generated by a monic irreducible polynomial of degree n. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: #x5C;mathbb#x7B;F#x7D;#x5F;#x7B;pⁿ#x7D;#x5C;cong#x7B;#x5C;mathbb#x7B;F#x7D;#x5F;p#x5B;x#x5D;#x5C;over(#x5C;hatf#x5F;n(x))#x7D;.",
"type": "example"
},
{
"text": "1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page,\nA field with a finite number of elements is called a Galois field.\nThe number of elements of the prime field k contained in a Galois field K is finite, and is therefore a natural prime p."
},
{
"ref": "2001, Joseph E. Bonin, A Brief Introduction To Matroid Theory, retrieved 2016-05-05",
"text": "The case of most interest to us will be that in which F is a finite field, the Galois field GF(q) for some prime power q. If q is prime, this field is #x5C;mathbb#x7B;Z#x7D;#x5F;q, the integers 0,1,#x5C;dots,q-1 with arithmetic modulo q.",
"type": "quotation"
},
{
"text": "2006, Debojyoti Battacharya, Debdeep Mukhopadhyay, D. RoyChowdhury, A Cellular Automata Based Approach for Generation of Large Primitive Polynomial and Its Application to RS-Coded MPSK Modulation, Samira El Yacoubi, Bastien Chopard, Stefania Bandini (editors), Cellular Automata: 7th International Conference, Proceedings, Springer, LNCS 4173, page 204,\nGeneration of large primitive polynomial over a Galois field has been a topic of intense research over the years. The problem of finding a primitive polynomial over a Galois field of a large degree is computationaly expensive and there is no deterministic algorithm for the same."
}
],
"glosses": [
"A finite field; a field that contains a finite number of elements."
],
"links": [
[
"algebra",
"algebra"
],
[
"finite field",
"finite field"
],
[
"finite",
"finite"
],
[
"elements",
"elements"
]
],
"raw_glosses": [
"(algebra) A finite field; a field that contains a finite number of elements."
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"wikipedia": [
"Finite field"
]
}
],
"word": "Galois field"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-01 from the enwiktionary dump dated 2024-04-21 using wiktextract (f4fd8c9 and c9440ce). 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.