Galois field
Noun

Galois field (plural Galois fields)

  1. (algebra) A finite field.
    The Galois field \mathrm{GF}(p^n) has order p^n and characteristic p.

    The multiplicative subgroup of a Galois field is cyclic.
    A Galois field \mathbb{F}_{p^n} is isomorphic to the quotient of the polynomial ring \mathbb{F}_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: \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 k contained in a Galois field K is finite, and is therefore a natural prime p.
    • 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, Lecture Notes in Computer Science 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.



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