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Galois field - Wiktionary, the free dictionary

  • ️Thu May 05 2016

From Wiktionary, the free dictionary

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

Galois field (plural Galois fields)

  1. (algebra) A finite field; a field that contains a finite number of elements.

    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[1], 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.
  • 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).