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Etymology

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

Noun

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Galois field (plural Galois fields)

  1. (algebra) A finite field; a field that contains a finite number of elements.
    The Galois field   has order   and characteristic  .
    The Galois field   is a finite extension of the Galois field   and the degree of the extension is  .
    The multiplicative subgroup of a Galois field is cyclic.
    A Galois field   is isomorphic to the quotient of the polynomial ring   adjoin   over the ideal generated by a monic irreducible polynomial of degree  . Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols:  .
    • 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   contained in a Galois field   is finite, and is therefore a natural prime  .
    • 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  , the integers   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

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  • For a given order, if a Galois field exists, it is unique, up to isomorphism.
  • Generally denoted   (but sometimes  ), where   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   or, more jocularly,   (pun intended).

Hypernyms

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Further reading

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