Galois field

English

Etymology

Named after French mathematician Évariste Galois (1811–1832).

Noun

Galois field (plural Galois fields)

(algebra) A finite field.
The Galois field ${\\displaystyle \\mathrm {GF} (p^{n})}$ has order ${\\displaystyle p^{n}}$ and characteristic ${\\displaystyle p}$ .
The Galois field ${\\displaystyle \\mathrm {GF} (p^{n})}$ is a finite extension of the Galois field ${\\displaystyle \\mathrm {GF} (p)}$ and the degree of the extension is ${\\displaystyle n}$ .
The multiplicative subgroup of a Galois field is cyclic.
A Galois field ${\\displaystyle \\mathbb {F} _{p^{n}}}$ is isomorphic to the quotient of the polynomial ring ${\\displaystyle \\mathbb {F} _{p}}$ adjoin ${\\displaystyle x}$ over the ideal generated by a monic irreducible polynomial of degree ${\\displaystyle n}$ . Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: ${\\displaystyle \\mathbb {F} _{p^{n}}\\cong {\\mathbb {F} _{p}[x] \\over ({\\hat {f}}_{n}(x))}}$ .
- 1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page,
  A field with a finite number of elements is called a Galois field.
  The number of elements of the prime field ${\\displaystyle k}$ contained in a Galois field ${\\displaystyle K}$ is finite, and is therefore a natural prime ${\\displaystyle p}$ .
- 2001, Joseph E. Bonin, A Brief Introduction To Matroid Theory‎, retrieved 2016-05-05:
  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 ${\\displaystyle \\mathbb {Z} _{q}}$ , the integers ${\\displaystyle 0,1,\\dots ,q-1}$ with arithmetic modulo q.
- 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,
  Generation 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^[sic] expensive and there is no deterministic algorithm for the same.

Usage notes

For a given order, if a Galois field exists, it is unique, up to isomorphism.
Generally denoted ${\\displaystyle \\mathrm {GF} (n)}$ (but sometimes ${\\displaystyle \\mathbb {F} _{n}}$ ), where ${\\displaystyle n}$ is the number of elements, which must be a positive integer power of a prime.
Although, strictly speaking, the "field of one element" does not exist (it is not a field in classical algebra), it is occasionally discussed in terms of how it might be meaningfully defined. Were it a meaningful concept, it would be a Galois field. It may be denoted ${\\displaystyle \\mathbb {F} _{1}}$ or, more jocularly, ${\\displaystyle \\mathbb {F} _{un}}$ (pun intended).

Hypernyms

perfect field

单词	Galois field
释义	Galois field English Etymology Named after French mathematician Évariste Galois (1811–1832). Noun Galois field (plural Galois fields) (algebra) A finite field. The Galois field* ${\\displaystyle \\mathrm {GF} (p^{n})}$ has order ${\\displaystyle p^{n}}$ and characteristic ${\\displaystyle p}$ .* The Galois field ${\\displaystyle \\mathrm {GF} (p^{n})}$ is a finite extension of the Galois field ${\\displaystyle \\mathrm {GF} (p)}$ and the degree of the extension is ${\\displaystyle n}$ . The multiplicative subgroup of a Galois field* is cyclic.* A Galois field ${\\displaystyle \\mathbb {F} _{p^{n}}}$ is isomorphic to the quotient of the polynomial ring ${\\displaystyle \\mathbb {F} _{p}}$ adjoin ${\\displaystyle x}$ over the ideal generated by a monic irreducible polynomial of degree ${\\displaystyle n}$ . Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: ${\\displaystyle \\mathbb {F} _{p^{n}}\\cong {\\mathbb {F} _{p}[x] \\over ({\\hat {f}}_{n}(x))}}$ . 1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page, A field with a finite number of elements is called a Galois field. The number of elements of the prime field ${\\displaystyle k}$ contained in a Galois field ${\\displaystyle K}$ is finite, and is therefore a natural prime ${\\displaystyle p}$ . 2001, Joseph E. Bonin, A Brief Introduction To Matroid Theory‎, retrieved 2016-05-05: 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 ${\\displaystyle \\mathbb {Z} _{q}}$ , the integers ${\\displaystyle 0,1,\\dots ,q-1}$ with arithmetic modulo q. 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, Generation 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^[sic] expensive and there is no deterministic algorithm for the same. Usage notes For a given order, if a Galois field exists, it is unique, up to isomorphism. Generally denoted ${\\displaystyle \\mathrm {GF} (n)}$ (but sometimes ${\\displaystyle \\mathbb {F} _{n}}$ ), where ${\\displaystyle n}$ is the number of elements, which must be a positive integer power of a prime. Although, strictly speaking, the "field of one element" does not exist (it is not a field in classical algebra), it is occasionally discussed in terms of how it might be meaningfully defined. Were it a meaningful concept, it would be a Galois field. It may be denoted ${\\displaystyle \\mathbb {F} _{1}}$ or, more jocularly, ${\\displaystyle \\mathbb {F} _{un}}$ (pun intended). Hypernyms perfect field Further reading Finite field arithmetic on Wikipedia.Wikipedia Finite ring on Wikipedia.Wikipedia Finite group on Wikipedia.Wikipedia Field with one element on Wikipedia.Wikipedia Galois geometry on Wikipedia.Wikipedia Galois theory on Wikipedia.Wikipedia Hamming space on Wikipedia.Wikipedia Galois field on Encyclopedia of Mathematics Galois field structure on Encyclopedia of Mathematics Finite Field on Wolfram MathWorld
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Galois field

English

Etymology

Noun

Usage notes

Hypernyms

Further reading